id	sid	tid	token	lemma	pos
ap-1406	1	1	acta	acta	PROPN
ap-1406	1	2	polytechnica	polytechnica	PROPN
ap-1406	1	3	vol	vol	NOUN
ap-1406	1	4	.	.	PUNCT
ap-1406	2	1	51	51	NUM
ap-1406	2	2	no	no	NOUN
ap-1406	2	3	.	.	PUNCT
ap-1406	3	1	4/2011	4/2011	NUM
ap-1406	3	2	ito	ito	PROPN
ap-1406	3	3	-	-	PUNCT
ap-1406	3	4	sadahiro	sadahiro	PROPN
ap-1406	3	5	numbers	number	NOUN
ap-1406	3	6	vs.	vs.	ADP
ap-1406	3	7	parry	parry	PROPN
ap-1406	3	8	numbers	number	NOUN
ap-1406	3	9	z.	z.	PROPN
ap-1406	3	10	masáková	masáková	PROPN
ap-1406	3	11	,	,	PUNCT
ap-1406	3	12	e.	e.	PROPN
ap-1406	3	13	pelantová	pelantová	PROPN
ap-1406	4	1	abstract	abstract	ADJ
ap-1406	4	2	we	we	PRON
ap-1406	4	3	consider	consider	VERB
ap-1406	4	4	a	a	DET
ap-1406	4	5	positional	positional	ADJ
ap-1406	4	6	numeration	numeration	NOUN
ap-1406	4	7	system	system	NOUN
ap-1406	4	8	with	with	ADP
ap-1406	4	9	a	a	DET
ap-1406	4	10	negative	negative	ADJ
ap-1406	4	11	base	base	NOUN
ap-1406	4	12	,	,	PUNCT
ap-1406	4	13	as	as	SCONJ
ap-1406	4	14	introduced	introduce	VERB
ap-1406	4	15	by	by	ADP
ap-1406	4	16	ito	ito	PROPN
ap-1406	4	17	and	and	CCONJ
ap-1406	4	18	sadahiro	sadahiro	PROPN
ap-1406	4	19	.	.	PUNCT
ap-1406	5	1	in	in	ADP
ap-1406	5	2	particular	particular	ADJ
ap-1406	5	3	,	,	PUNCT
ap-1406	5	4	we	we	PRON
ap-1406	5	5	focus	focus	VERB
ap-1406	5	6	on	on	ADP
ap-1406	5	7	the	the	DET
ap-1406	5	8	algebraic	algebraic	ADJ
ap-1406	5	9	properties	property	NOUN
ap-1406	5	10	of	of	ADP
ap-1406	5	11	negative	negative	ADJ
ap-1406	5	12	bases	basis	NOUN
ap-1406	5	13	−β	−β	NOUN
ap-1406	5	14	for	for	ADP
ap-1406	5	15	which	which	PRON
ap-1406	5	16	the	the	DET
ap-1406	5	17	corresponding	corresponding	ADJ
ap-1406	5	18	dynamical	dynamical	ADJ
ap-1406	5	19	system	system	NOUN
ap-1406	5	20	is	be	AUX
ap-1406	5	21	sofic	sofic	ADJ
ap-1406	5	22	,	,	PUNCT
ap-1406	5	23	which	which	PRON
ap-1406	5	24	happens	happen	VERB
ap-1406	5	25	,	,	PUNCT
ap-1406	5	26	according	accord	VERB
ap-1406	5	27	to	to	ADP
ap-1406	5	28	ito	ito	PROPN
ap-1406	5	29	and	and	CCONJ
ap-1406	5	30	sadahiro	sadahiro	PROPN
ap-1406	5	31	,	,	PUNCT
ap-1406	5	32	if	if	SCONJ
ap-1406	5	33	and	and	CCONJ
ap-1406	5	34	only	only	ADV
ap-1406	5	35	if	if	SCONJ
ap-1406	5	36	the	the	DET
ap-1406	5	37	(	(	PUNCT
ap-1406	5	38	−β)-expansion	−β)-expansion	NOUN
ap-1406	5	39	of	of	ADP
ap-1406	5	40	−	−	PROPN
ap-1406	5	41	β	β	NOUN
ap-1406	5	42	β	β	NOUN
ap-1406	5	43	+	+	CCONJ
ap-1406	5	44	1	1	NUM
ap-1406	5	45	is	be	AUX
ap-1406	5	46	eventually	eventually	ADV
ap-1406	5	47	periodic	periodic	ADJ
ap-1406	5	48	.	.	PUNCT
ap-1406	6	1	we	we	PRON
ap-1406	6	2	call	call	VERB
ap-1406	6	3	such	such	ADJ
ap-1406	6	4	numbers	number	NOUN
ap-1406	6	5	β	β	PROPN
ap-1406	6	6	ito	ito	PROPN
ap-1406	6	7	-	-	PROPN
ap-1406	6	8	sadahiro	sadahiro	PROPN
ap-1406	6	9	numbers	number	NOUN
ap-1406	6	10	,	,	PUNCT
ap-1406	6	11	and	and	CCONJ
ap-1406	6	12	we	we	PRON
ap-1406	6	13	compare	compare	VERB
ap-1406	6	14	their	their	PRON
ap-1406	6	15	properties	property	NOUN
ap-1406	6	16	with	with	ADP
ap-1406	6	17	those	those	PRON
ap-1406	6	18	of	of	ADP
ap-1406	6	19	parry	parry	PROPN
ap-1406	6	20	numbers	number	NOUN
ap-1406	6	21	,	,	PUNCT
ap-1406	6	22	which	which	PRON
ap-1406	6	23	occur	occur	VERB
ap-1406	6	24	in	in	ADP
ap-1406	6	25	the	the	DET
ap-1406	6	26	same	same	ADJ
ap-1406	6	27	context	context	NOUN
ap-1406	6	28	for	for	ADP
ap-1406	6	29	the	the	DET
ap-1406	6	30	rényi	rényi	PROPN
ap-1406	6	31	positive	positive	ADJ
ap-1406	6	32	base	base	NOUN
ap-1406	6	33	numeration	numeration	NOUN
ap-1406	6	34	system	system	NOUN
ap-1406	6	35	.	.	PUNCT
ap-1406	7	1	keywords	keyword	NOUN
ap-1406	7	2	:	:	PUNCT
ap-1406	7	3	numeration	numeration	NOUN
ap-1406	7	4	systems	system	NOUN
ap-1406	7	5	,	,	PUNCT
ap-1406	7	6	negative	negative	ADJ
ap-1406	7	7	base	base	NOUN
ap-1406	7	8	,	,	PUNCT
ap-1406	7	9	pisot	pisot	ADJ
ap-1406	7	10	number	number	NOUN
ap-1406	7	11	,	,	PUNCT
ap-1406	7	12	parry	parry	VERB
ap-1406	7	13	number	number	NOUN
ap-1406	7	14	.	.	PUNCT
ap-1406	8	1	1	1	NUM
ap-1406	8	2	introduction	introduction	NOUN
ap-1406	8	3	the	the	DET
ap-1406	8	4	expansion	expansion	NOUN
ap-1406	8	5	of	of	ADP
ap-1406	8	6	a	a	DET
ap-1406	8	7	real	real	ADJ
ap-1406	8	8	number	number	NOUN
ap-1406	8	9	in	in	ADP
ap-1406	8	10	the	the	DET
ap-1406	8	11	positional	positional	ADJ
ap-1406	8	12	number	number	NOUN
ap-1406	8	13	system	system	NOUN
ap-1406	8	14	with	with	ADP
ap-1406	8	15	base	base	NOUN
ap-1406	8	16	β	β	X
ap-1406	8	17	>	>	X
ap-1406	8	18	1	1	NUM
ap-1406	8	19	,	,	PUNCT
ap-1406	8	20	as	as	SCONJ
ap-1406	8	21	defined	define	VERB
ap-1406	8	22	by	by	ADP
ap-1406	8	23	rényi	rényi	PROPN
ap-1406	9	1	[	[	X
ap-1406	9	2	12	12	NUM
ap-1406	9	3	]	]	PUNCT
ap-1406	9	4	is	be	AUX
ap-1406	9	5	closely	closely	ADV
ap-1406	9	6	related	relate	VERB
ap-1406	9	7	to	to	ADP
ap-1406	9	8	the	the	DET
ap-1406	9	9	transformation	transformation	NOUN
ap-1406	9	10	t	t	NOUN
ap-1406	9	11	:	:	PUNCT
ap-1406	10	1	[	[	X
ap-1406	10	2	0	0	NUM
ap-1406	10	3	,	,	PUNCT
ap-1406	10	4	1	1	NUM
ap-1406	10	5	)	)	PUNCT
ap-1406	10	6	�	�	PROPN
ap-1406	10	7	→	→	SYM
ap-1406	11	1	[	[	X
ap-1406	11	2	0	0	NUM
ap-1406	11	3	,	,	PUNCT
ap-1406	11	4	1	1	NUM
ap-1406	11	5	)	)	PUNCT
ap-1406	11	6	,	,	PUNCT
ap-1406	11	7	given	give	VERB
ap-1406	11	8	by	by	ADP
ap-1406	11	9	the	the	DET
ap-1406	11	10	prescription	prescription	NOUN
ap-1406	11	11	t	t	NOUN
ap-1406	11	12	(	(	PUNCT
ap-1406	11	13	x	x	NOUN
ap-1406	11	14	)	)	PUNCT
ap-1406	11	15	:	:	PUNCT
ap-1406	11	16	=	=	PUNCT
ap-1406	11	17	βx	βx	ADP
ap-1406	11	18	−	−	NOUN
ap-1406	11	19	#	#	NOUN
ap-1406	11	20	βx$.	βx$.	NOUN
ap-1406	11	21	every	every	DET
ap-1406	11	22	x	x	SYM
ap-1406	11	23	∈	∈	PROPN
ap-1406	12	1	[	[	X
ap-1406	12	2	0	0	NUM
ap-1406	12	3	,	,	PUNCT
ap-1406	12	4	1	1	NUM
ap-1406	12	5	)	)	PUNCT
ap-1406	12	6	is	be	AUX
ap-1406	12	7	a	a	DET
ap-1406	12	8	sum	sum	NOUN
ap-1406	12	9	of	of	ADP
ap-1406	12	10	the	the	DET
ap-1406	12	11	infinite	infinite	ADJ
ap-1406	12	12	series	series	NOUN
ap-1406	12	13	x	x	X
ap-1406	13	1	=	=	PUNCT
ap-1406	13	2	∞∑	∞∑	NOUN
ap-1406	13	3	i=1	i=1	X
ap-1406	13	4	xi	xi	X
ap-1406	13	5	βi	βi	PROPN
ap-1406	13	6	,	,	PUNCT
ap-1406	13	7	where	where	SCONJ
ap-1406	13	8	xi	xi	X
ap-1406	13	9	=	=	NOUN
ap-1406	13	10	#	#	SYM
ap-1406	13	11	βt	βt	NOUN
ap-1406	13	12	i−1(x)$	i−1(x)$	NOUN
ap-1406	13	13	(	(	PUNCT
ap-1406	13	14	1	1	NUM
ap-1406	13	15	)	)	PUNCT
ap-1406	13	16	for	for	ADP
ap-1406	13	17	i	i	PRON
ap-1406	13	18	=	=	SYM
ap-1406	13	19	1	1	NUM
ap-1406	13	20	,	,	PUNCT
ap-1406	13	21	2	2	NUM
ap-1406	13	22	,	,	PUNCT
ap-1406	13	23	3	3	NUM
ap-1406	13	24	,	,	PUNCT
ap-1406	13	25	.	.	PUNCT
ap-1406	13	26	.	.	PUNCT
ap-1406	13	27	.	.	PUNCT
ap-1406	14	1	directly	directly	ADV
ap-1406	14	2	from	from	ADP
ap-1406	14	3	the	the	DET
ap-1406	14	4	definition	definition	NOUN
ap-1406	14	5	of	of	ADP
ap-1406	14	6	the	the	DET
ap-1406	14	7	transformation	transformation	NOUN
ap-1406	14	8	t	t	NOUN
ap-1406	14	9	we	we	PRON
ap-1406	14	10	can	can	AUX
ap-1406	14	11	derive	derive	VERB
ap-1406	14	12	that	that	SCONJ
ap-1406	14	13	the	the	DET
ap-1406	14	14	‘	'	PUNCT
ap-1406	14	15	digits	digit	NOUN
ap-1406	14	16	’	'	PUNCT
ap-1406	14	17	xi	xi	AUX
ap-1406	14	18	take	take	VERB
ap-1406	14	19	values	value	NOUN
ap-1406	14	20	in	in	ADP
ap-1406	14	21	the	the	DET
ap-1406	14	22	set	set	NOUN
ap-1406	14	23	{	{	PUNCT
ap-1406	14	24	0	0	NUM
ap-1406	14	25	,	,	PUNCT
ap-1406	14	26	1	1	NUM
ap-1406	14	27	,	,	PUNCT
ap-1406	14	28	2	2	NUM
ap-1406	14	29	,	,	PUNCT
ap-1406	14	30	.	.	PUNCT
ap-1406	14	31	.	.	PUNCT
ap-1406	15	1	.	.	PUNCT
ap-1406	16	1	,	,	PUNCT
ap-1406	16	2	%	%	INTJ
ap-1406	16	3	β	β	X
ap-1406	16	4	&	&	CCONJ
ap-1406	16	5	−	−	PROPN
ap-1406	16	6	1	1	NUM
ap-1406	16	7	}	}	PUNCT
ap-1406	16	8	for	for	ADP
ap-1406	16	9	i	i	PROPN
ap-1406	16	10	=	=	NOUN
ap-1406	16	11	1	1	NUM
ap-1406	16	12	,	,	PUNCT
ap-1406	16	13	2	2	NUM
ap-1406	16	14	,	,	PUNCT
ap-1406	16	15	3	3	NUM
ap-1406	16	16	,	,	PUNCT
ap-1406	16	17	.	.	PUNCT
ap-1406	16	18	.	.	PUNCT
ap-1406	17	1	..	..	PUNCT
ap-1406	18	1	the	the	DET
ap-1406	18	2	expression	expression	NOUN
ap-1406	18	3	of	of	ADP
ap-1406	18	4	x	x	PUNCT
ap-1406	18	5	in	in	ADP
ap-1406	18	6	the	the	DET
ap-1406	18	7	form	form	NOUN
ap-1406	18	8	(	(	PUNCT
ap-1406	18	9	1	1	X
ap-1406	18	10	)	)	PUNCT
ap-1406	18	11	is	be	AUX
ap-1406	18	12	called	call	VERB
ap-1406	18	13	the	the	DET
ap-1406	18	14	β	β	NOUN
ap-1406	18	15	-	-	NOUN
ap-1406	18	16	expansion	expansion	NOUN
ap-1406	18	17	of	of	ADP
ap-1406	18	18	x.	x.	NOUN
ap-1406	18	19	the	the	DET
ap-1406	18	20	number	number	NOUN
ap-1406	18	21	x	x	PUNCT
ap-1406	18	22	is	be	AUX
ap-1406	18	23	thus	thus	ADV
ap-1406	18	24	represented	represent	VERB
ap-1406	18	25	by	by	ADP
ap-1406	18	26	the	the	DET
ap-1406	18	27	infinite	infinite	ADJ
ap-1406	18	28	word	word	NOUN
ap-1406	18	29	dβ(x	dβ(x	NOUN
ap-1406	18	30	)	)	PUNCT
ap-1406	18	31	=	=	SYM
ap-1406	18	32	x1x2x3	x1x2x3	PROPN
ap-1406	18	33	.	.	PUNCT
ap-1406	18	34	.	.	PUNCT
ap-1406	18	35	.	.	PUNCT
ap-1406	19	1	∈	∈	PROPN
ap-1406	20	1	an	an	PRON
ap-1406	20	2	over	over	ADP
ap-1406	20	3	the	the	DET
ap-1406	20	4	alphabet	alphabet	NOUN
ap-1406	20	5	a	a	X
ap-1406	20	6	=	=	X
ap-1406	20	7	{	{	PUNCT
ap-1406	20	8	0	0	NUM
ap-1406	20	9	,	,	PUNCT
ap-1406	20	10	1	1	NUM
ap-1406	20	11	,	,	PUNCT
ap-1406	20	12	2	2	NUM
ap-1406	20	13	,	,	PUNCT
ap-1406	20	14	.	.	PUNCT
ap-1406	20	15	.	.	PUNCT
ap-1406	20	16	.	.	PUNCT
ap-1406	21	1	,	,	PUNCT
ap-1406	21	2	%	%	INTJ
ap-1406	21	3	β	β	X
ap-1406	21	4	&	&	CCONJ
ap-1406	21	5	−	−	PROPN
ap-1406	21	6	1	1	NUM
ap-1406	21	7	}	}	PUNCT
ap-1406	21	8	.	.	PUNCT
ap-1406	22	1	from	from	ADP
ap-1406	22	2	the	the	DET
ap-1406	22	3	definition	definition	NOUN
ap-1406	22	4	of	of	ADP
ap-1406	22	5	the	the	DET
ap-1406	22	6	transformation	transformation	NOUN
ap-1406	22	7	t	t	NOUN
ap-1406	22	8	we	we	PRON
ap-1406	22	9	can	can	AUX
ap-1406	22	10	derive	derive	VERB
ap-1406	22	11	another	another	DET
ap-1406	22	12	important	important	ADJ
ap-1406	22	13	property	property	NOUN
ap-1406	22	14	,	,	PUNCT
ap-1406	22	15	namely	namely	ADV
ap-1406	22	16	that	that	SCONJ
ap-1406	22	17	the	the	DET
ap-1406	22	18	ordering	ordering	NOUN
ap-1406	22	19	on	on	ADP
ap-1406	22	20	real	real	ADJ
ap-1406	22	21	numbers	number	NOUN
ap-1406	22	22	is	be	AUX
ap-1406	22	23	carried	carry	VERB
ap-1406	22	24	over	over	ADP
ap-1406	22	25	to	to	ADP
ap-1406	22	26	the	the	DET
ap-1406	22	27	ordering	ordering	NOUN
ap-1406	22	28	of	of	ADP
ap-1406	22	29	β	β	NOUN
ap-1406	22	30	-	-	NOUN
ap-1406	22	31	expansions	expansion	NOUN
ap-1406	22	32	.	.	PUNCT
ap-1406	23	1	in	in	ADP
ap-1406	23	2	particular	particular	ADJ
ap-1406	23	3	,	,	PUNCT
ap-1406	23	4	we	we	PRON
ap-1406	23	5	have	have	VERB
ap-1406	23	6	for	for	ADP
ap-1406	23	7	x	x	X
ap-1406	23	8	,	,	PUNCT
ap-1406	23	9	y	y	PROPN
ap-1406	23	10	∈	∈	PROPN
ap-1406	24	1	[	[	X
ap-1406	24	2	0	0	NUM
ap-1406	24	3	,	,	PUNCT
ap-1406	24	4	1	1	NUM
ap-1406	24	5	)	)	PUNCT
ap-1406	24	6	that	that	SCONJ
ap-1406	24	7	x	x	X
ap-1406	24	8	≤	≤	ADJ
ap-1406	24	9	y	y	NUM
ap-1406	24	10	⇐	⇐	ADJ
ap-1406	24	11	⇒	⇒	PROPN
ap-1406	24	12	dβ(x	dβ(x	NOUN
ap-1406	24	13	)	)	PUNCT
ap-1406	24	14	)	)	PUNCT
ap-1406	25	1	dβ(y	dβ(y	NOUN
ap-1406	25	2	)	)	PUNCT
ap-1406	25	3	,	,	PUNCT
ap-1406	25	4	where	where	SCONJ
ap-1406	25	5	)	)	PUNCT
ap-1406	25	6	is	be	AUX
ap-1406	25	7	the	the	DET
ap-1406	25	8	lexicographical	lexicographical	ADJ
ap-1406	25	9	order	order	NOUN
ap-1406	25	10	on	on	ADP
ap-1406	25	11	an	an	PRON
ap-1406	25	12	,	,	PUNCT
ap-1406	25	13	(	(	PUNCT
ap-1406	25	14	ordering	order	VERB
ap-1406	25	15	on	on	ADP
ap-1406	25	16	the	the	DET
ap-1406	25	17	alphabet	alphabet	NOUN
ap-1406	25	18	a	a	PRON
ap-1406	25	19	is	be	AUX
ap-1406	25	20	usual	usual	ADJ
ap-1406	25	21	,	,	PUNCT
ap-1406	26	1	0	0	PUNCT
ap-1406	26	2	<	<	X
ap-1406	26	3	1	1	NUM
ap-1406	26	4	<	<	X
ap-1406	26	5	2	2	NUM
ap-1406	26	6	<	<	X
ap-1406	26	7	.	.	PUNCT
ap-1406	26	8	.	.	PUNCT
ap-1406	26	9	.	.	PUNCT
ap-1406	27	1	<	<	X
ap-1406	28	1	%	%	X
ap-1406	28	2	β	β	X
ap-1406	28	3	&	&	CCONJ
ap-1406	28	4	−	−	PROPN
ap-1406	28	5	1	1	NUM
ap-1406	28	6	)	)	PUNCT
ap-1406	28	7	.	.	PUNCT
ap-1406	29	1	in	in	ADP
ap-1406	29	2	[	[	X
ap-1406	29	3	11	11	NUM
ap-1406	29	4	]	]	PUNCT
ap-1406	29	5	,	,	PUNCT
ap-1406	29	6	parry	parry	PROPN
ap-1406	29	7	has	have	AUX
ap-1406	29	8	provided	provide	VERB
ap-1406	29	9	a	a	DET
ap-1406	29	10	criterion	criterion	NOUN
ap-1406	29	11	which	which	PRON
ap-1406	29	12	decides	decide	VERB
ap-1406	29	13	whether	whether	SCONJ
ap-1406	29	14	an	an	DET
ap-1406	29	15	infinite	infinite	ADJ
ap-1406	29	16	word	word	NOUN
ap-1406	29	17	in	in	ADP
ap-1406	29	18	an	an	DET
ap-1406	29	19	is	is	NOUN
ap-1406	29	20	or	or	CCONJ
ap-1406	29	21	not	not	PART
ap-1406	29	22	a	a	DET
ap-1406	29	23	βexpansion	βexpansion	NOUN
ap-1406	29	24	of	of	ADP
ap-1406	29	25	some	some	DET
ap-1406	29	26	real	real	ADJ
ap-1406	29	27	number	number	NOUN
ap-1406	29	28	x.	x.	NOUN
ap-1406	30	1	the	the	DET
ap-1406	30	2	criterion	criterion	NOUN
ap-1406	30	3	is	be	AUX
ap-1406	30	4	formulated	formulate	VERB
ap-1406	30	5	using	use	VERB
ap-1406	30	6	the	the	DET
ap-1406	30	7	so	so	ADV
ap-1406	30	8	-	-	PUNCT
ap-1406	30	9	called	call	VERB
ap-1406	30	10	infinite	infinite	ADJ
ap-1406	30	11	expansion	expansion	NOUN
ap-1406	30	12	of	of	ADP
ap-1406	30	13	1	1	NUM
ap-1406	30	14	,	,	PUNCT
ap-1406	30	15	denoted	denote	VERB
ap-1406	30	16	by	by	ADP
ap-1406	30	17	d∗β(1	d∗β(1	NOUN
ap-1406	30	18	)	)	PUNCT
ap-1406	30	19	,	,	PUNCT
ap-1406	30	20	defined	define	VERB
ap-1406	30	21	as	as	ADP
ap-1406	30	22	a	a	DET
ap-1406	30	23	limit	limit	NOUN
ap-1406	30	24	in	in	ADP
ap-1406	30	25	the	the	DET
ap-1406	30	26	space	space	NOUN
ap-1406	30	27	an	an	DET
ap-1406	30	28	equipped	equip	VERB
ap-1406	30	29	with	with	ADP
ap-1406	30	30	the	the	DET
ap-1406	30	31	product	product	NOUN
ap-1406	30	32	topology	topology	NOUN
ap-1406	30	33	,	,	PUNCT
ap-1406	30	34	by	by	ADP
ap-1406	30	35	d∗β(1	d∗β(1	NOUN
ap-1406	30	36	)	)	PUNCT
ap-1406	30	37	:	:	PUNCT
ap-1406	30	38	=	=	PUNCT
ap-1406	30	39	lim	lim	NOUN
ap-1406	30	40	ε→0	ε→0	VERB
ap-1406	30	41	+	+	CCONJ
ap-1406	30	42	dβ(1	dβ(1	PROPN
ap-1406	30	43	−	−	PROPN
ap-1406	30	44	ε	ε	PROPN
ap-1406	30	45	)	)	PUNCT
ap-1406	30	46	.	.	PUNCT
ap-1406	31	1	according	accord	VERB
ap-1406	31	2	to	to	ADP
ap-1406	31	3	parry	parry	PROPN
ap-1406	31	4	,	,	PUNCT
ap-1406	31	5	the	the	DET
ap-1406	31	6	string	string	NOUN
ap-1406	31	7	x1x2x3	x1x2x3	PROPN
ap-1406	31	8	.	.	PUNCT
ap-1406	31	9	.	.	PUNCT
ap-1406	31	10	.	.	PUNCT
ap-1406	32	1	∈	∈	PROPN
ap-1406	32	2	an	an	DET
ap-1406	32	3	represents	represent	VERB
ap-1406	32	4	the	the	DET
ap-1406	32	5	β	β	NOUN
ap-1406	32	6	-	-	NOUN
ap-1406	32	7	expansion	expansion	NOUN
ap-1406	32	8	of	of	ADP
ap-1406	32	9	a	a	DET
ap-1406	32	10	number	number	NOUN
ap-1406	32	11	x	x	SYM
ap-1406	32	12	∈	∈	PROPN
ap-1406	33	1	[	[	X
ap-1406	33	2	0	0	NUM
ap-1406	33	3	,	,	PUNCT
ap-1406	33	4	1	1	NUM
ap-1406	33	5	)	)	PUNCT
ap-1406	34	1	if	if	SCONJ
ap-1406	34	2	and	and	CCONJ
ap-1406	34	3	only	only	ADV
ap-1406	34	4	if	if	SCONJ
ap-1406	34	5	xixi+1xi+2	xixi+1xi+2	PROPN
ap-1406	34	6	.	.	PUNCT
ap-1406	34	7	.	.	PUNCT
ap-1406	35	1	.	.	PUNCT
ap-1406	36	1	≺	≺	NOUN
ap-1406	36	2	d∗β(1	d∗β(1	NOUN
ap-1406	36	3	)	)	PUNCT
ap-1406	36	4	(	(	PUNCT
ap-1406	36	5	2	2	X
ap-1406	36	6	)	)	PUNCT
ap-1406	36	7	for	for	ADP
ap-1406	36	8	every	every	DET
ap-1406	36	9	i	i	NOUN
ap-1406	36	10	=	=	NOUN
ap-1406	36	11	1	1	NUM
ap-1406	36	12	,	,	PUNCT
ap-1406	36	13	2	2	NUM
ap-1406	36	14	,	,	PUNCT
ap-1406	36	15	3	3	NUM
ap-1406	36	16	,	,	PUNCT
ap-1406	36	17	.	.	PUNCT
ap-1406	36	18	.	.	PUNCT
ap-1406	36	19	.	.	PUNCT
ap-1406	37	1	condition	condition	NOUN
ap-1406	37	2	(	(	PUNCT
ap-1406	37	3	2	2	X
ap-1406	37	4	)	)	PUNCT
ap-1406	37	5	ensures	ensure	VERB
ap-1406	37	6	that	that	SCONJ
ap-1406	37	7	the	the	DET
ap-1406	37	8	set	set	NOUN
ap-1406	37	9	dβ	dβ	NOUN
ap-1406	37	10	=	=	PUNCT
ap-1406	37	11	{	{	PUNCT
ap-1406	37	12	dβ(x	dβ(x	NOUN
ap-1406	37	13	)	)	PUNCT
ap-1406	37	14	|	|	ADV
ap-1406	37	15	x	x	SYM
ap-1406	37	16	∈	∈	PROPN
ap-1406	38	1	[	[	X
ap-1406	38	2	0	0	NUM
ap-1406	38	3	,	,	PUNCT
ap-1406	38	4	1	1	NUM
ap-1406	38	5	)	)	PUNCT
ap-1406	38	6	}	}	PUNCT
ap-1406	38	7	is	be	AUX
ap-1406	38	8	shift	shift	NOUN
ap-1406	38	9	invariant	invariant	ADJ
ap-1406	38	10	,	,	PUNCT
ap-1406	38	11	and	and	CCONJ
ap-1406	38	12	so	so	ADV
ap-1406	38	13	the	the	DET
ap-1406	38	14	closure	closure	NOUN
ap-1406	38	15	of	of	ADP
ap-1406	38	16	dβ	dβ	NOUN
ap-1406	38	17	in	in	ADP
ap-1406	38	18	an	an	PRON
ap-1406	38	19	,	,	PUNCT
ap-1406	38	20	denoted	denote	VERB
ap-1406	38	21	by	by	ADP
ap-1406	38	22	sβ	sβ	NOUN
ap-1406	38	23	,	,	PUNCT
ap-1406	38	24	is	be	AUX
ap-1406	38	25	a	a	DET
ap-1406	38	26	subshift	subshift	NOUN
ap-1406	38	27	of	of	ADP
ap-1406	38	28	the	the	DET
ap-1406	38	29	full	full	ADJ
ap-1406	38	30	shift	shift	NOUN
ap-1406	38	31	an	an	PRON
ap-1406	38	32	.	.	PUNCT
ap-1406	39	1	the	the	DET
ap-1406	39	2	notion	notion	NOUN
ap-1406	39	3	of	of	ADP
ap-1406	39	4	β	β	NOUN
ap-1406	39	5	-	-	NOUN
ap-1406	39	6	expansion	expansion	NOUN
ap-1406	39	7	can	can	AUX
ap-1406	39	8	naturally	naturally	ADV
ap-1406	39	9	be	be	AUX
ap-1406	39	10	extended	extend	VERB
ap-1406	39	11	to	to	ADP
ap-1406	39	12	all	all	DET
ap-1406	39	13	non	non	ADJ
ap-1406	39	14	-	-	ADJ
ap-1406	39	15	negative	negative	ADJ
ap-1406	39	16	real	real	ADJ
ap-1406	39	17	numbers	number	NOUN
ap-1406	39	18	:	:	PUNCT
ap-1406	39	19	the	the	DET
ap-1406	39	20	expression	expression	NOUN
ap-1406	39	21	of	of	ADP
ap-1406	39	22	a	a	DET
ap-1406	39	23	positive	positive	ADJ
ap-1406	39	24	real	real	ADJ
ap-1406	39	25	number	number	NOUN
ap-1406	39	26	y	y	PROPN
ap-1406	39	27	in	in	ADP
ap-1406	39	28	the	the	DET
ap-1406	39	29	form	form	NOUN
ap-1406	40	1	y	y	NOUN
ap-1406	40	2	=	=	PUNCT
ap-1406	40	3	ykβk	ykβk	NOUN
ap-1406	40	4	+	+	CCONJ
ap-1406	40	5	yk−1β	yk−1β	PROPN
ap-1406	40	6	k−1	k−1	PROPN
ap-1406	40	7	+	+	CCONJ
ap-1406	40	8	yk−2β	yk−2β	PROPN
ap-1406	40	9	k−2	k−2	PROPN
ap-1406	41	1	+	+	X
ap-1406	41	2	.	.	PUNCT
ap-1406	41	3	.	.	PUNCT
ap-1406	41	4	.	.	PUNCT
ap-1406	42	1	,	,	PUNCT
ap-1406	42	2	(	(	PUNCT
ap-1406	42	3	3	3	X
ap-1406	42	4	)	)	PUNCT
ap-1406	42	5	where	where	SCONJ
ap-1406	42	6	k	k	PROPN
ap-1406	42	7	∈	∈	PROPN
ap-1406	42	8	z	z	PROPN
ap-1406	42	9	and	and	CCONJ
ap-1406	42	10	ykyk−1yk−2	ykyk−1yk−2	PROPN
ap-1406	42	11	.	.	PUNCT
ap-1406	42	12	.	.	PUNCT
ap-1406	42	13	.	.	PUNCT
ap-1406	43	1	∈	∈	PROPN
ap-1406	43	2	dβ	dβ	ADP
ap-1406	43	3	,	,	PUNCT
ap-1406	43	4	is	be	AUX
ap-1406	43	5	called	call	VERB
ap-1406	43	6	the	the	DET
ap-1406	43	7	β	β	NOUN
ap-1406	43	8	-	-	NOUN
ap-1406	43	9	expansion	expansion	NOUN
ap-1406	43	10	of	of	ADP
ap-1406	43	11	y.	y.	PROPN
ap-1406	43	12	real	real	ADJ
ap-1406	43	13	numbers	number	NOUN
ap-1406	43	14	y	y	PROPN
ap-1406	43	15	having	have	VERB
ap-1406	43	16	in	in	ADP
ap-1406	43	17	the	the	DET
ap-1406	43	18	β	β	NOUN
ap-1406	43	19	-	-	NOUN
ap-1406	43	20	expansion	expansion	NOUN
ap-1406	43	21	of	of	ADP
ap-1406	43	22	|y|	|y|	ADJ
ap-1406	43	23	vanishing	vanishing	ADJ
ap-1406	43	24	digits	digit	NOUN
ap-1406	43	25	yi	yi	PROPN
ap-1406	43	26	for	for	ADP
ap-1406	43	27	all	all	DET
ap-1406	43	28	i	i	PRON
ap-1406	43	29	<	<	X
ap-1406	43	30	0	0	NUM
ap-1406	43	31	are	be	AUX
ap-1406	43	32	usually	usually	ADV
ap-1406	43	33	called	call	VERB
ap-1406	43	34	βintegers	βinteger	NOUN
ap-1406	43	35	,	,	PUNCT
ap-1406	43	36	and	and	CCONJ
ap-1406	43	37	the	the	DET
ap-1406	43	38	set	set	NOUN
ap-1406	43	39	of	of	ADP
ap-1406	43	40	β	β	NOUN
ap-1406	43	41	-	-	NOUN
ap-1406	43	42	integers	integer	NOUN
ap-1406	43	43	is	be	AUX
ap-1406	43	44	denoted	denote	VERB
ap-1406	43	45	by	by	ADP
ap-1406	43	46	zβ	zβ	PROPN
ap-1406	43	47	.	.	PUNCT
ap-1406	44	1	the	the	DET
ap-1406	44	2	notion	notion	NOUN
ap-1406	44	3	of	of	ADP
ap-1406	44	4	β	β	NOUN
ap-1406	44	5	-	-	NOUN
ap-1406	44	6	integers	integer	NOUN
ap-1406	44	7	was	be	AUX
ap-1406	44	8	first	first	ADV
ap-1406	44	9	considered	consider	VERB
ap-1406	44	10	in	in	ADP
ap-1406	44	11	[	[	X
ap-1406	44	12	3	3	X
ap-1406	44	13	]	]	PUNCT
ap-1406	44	14	as	as	ADP
ap-1406	44	15	an	an	DET
ap-1406	44	16	aperiodic	aperiodic	ADJ
ap-1406	44	17	structure	structure	NOUN
ap-1406	44	18	modeling	model	VERB
ap-1406	44	19	non	non	ADJ
ap-1406	44	20	-	-	ADJ
ap-1406	44	21	crystallographic	crystallographic	ADJ
ap-1406	44	22	materials	material	NOUN
ap-1406	44	23	with	with	ADP
ap-1406	44	24	long	long	ADJ
ap-1406	44	25	range	range	NOUN
ap-1406	44	26	order	order	NOUN
ap-1406	44	27	,	,	PUNCT
ap-1406	44	28	called	call	VERB
ap-1406	44	29	quasicrystals	quasicrystal	NOUN
ap-1406	44	30	.	.	PUNCT
ap-1406	45	1	numbers	number	NOUN
ap-1406	45	2	y	y	PROPN
ap-1406	45	3	with	with	ADP
ap-1406	45	4	finitely	finitely	ADV
ap-1406	45	5	many	many	ADJ
ap-1406	45	6	non	non	ADJ
ap-1406	45	7	-	-	ADJ
ap-1406	45	8	zero	zero	ADJ
ap-1406	45	9	digits	digit	NOUN
ap-1406	45	10	in	in	ADP
ap-1406	45	11	the	the	DET
ap-1406	45	12	β	β	NOUN
ap-1406	45	13	-	-	NOUN
ap-1406	45	14	expansion	expansion	NOUN
ap-1406	45	15	of	of	ADP
ap-1406	45	16	|y|	|y|	PROPN
ap-1406	45	17	form	form	NOUN
ap-1406	45	18	the	the	DET
ap-1406	45	19	set	set	NOUN
ap-1406	45	20	denoted	denote	VERB
ap-1406	45	21	by	by	ADP
ap-1406	45	22	fin(β	fin(β	PROPN
ap-1406	45	23	)	)	PUNCT
ap-1406	45	24	.	.	PUNCT
ap-1406	46	1	the	the	DET
ap-1406	46	2	choice	choice	NOUN
ap-1406	46	3	of	of	ADP
ap-1406	46	4	the	the	DET
ap-1406	46	5	base	base	NOUN
ap-1406	46	6	β	β	X
ap-1406	46	7	>	>	X
ap-1406	46	8	1	1	NUM
ap-1406	46	9	strongly	strongly	ADV
ap-1406	46	10	influences	influence	VERB
ap-1406	46	11	the	the	DET
ap-1406	46	12	properties	property	NOUN
ap-1406	46	13	of	of	ADP
ap-1406	46	14	β	β	NOUN
ap-1406	46	15	-	-	NOUN
ap-1406	46	16	expansions	expansion	NOUN
ap-1406	46	17	.	.	PUNCT
ap-1406	47	1	it	it	PRON
ap-1406	47	2	turns	turn	VERB
ap-1406	47	3	out	out	ADP
ap-1406	47	4	that	that	SCONJ
ap-1406	47	5	an	an	DET
ap-1406	47	6	important	important	ADJ
ap-1406	47	7	role	role	NOUN
ap-1406	47	8	among	among	ADP
ap-1406	47	9	bases	basis	NOUN
ap-1406	47	10	is	be	AUX
ap-1406	47	11	played	play	VERB
ap-1406	47	12	by	by	ADP
ap-1406	47	13	such	such	ADJ
ap-1406	47	14	numbers	number	NOUN
ap-1406	47	15	β	β	X
ap-1406	47	16	for	for	ADP
ap-1406	47	17	which	which	PRON
ap-1406	47	18	d∗β(1	d∗β(1	NOUN
ap-1406	47	19	)	)	PUNCT
ap-1406	47	20	is	be	AUX
ap-1406	47	21	eventually	eventually	ADV
ap-1406	47	22	periodic	periodic	ADJ
ap-1406	47	23	.	.	PUNCT
ap-1406	48	1	parry	parry	VERB
ap-1406	48	2	himself	himself	PRON
ap-1406	48	3	called	call	VERB
ap-1406	48	4	these	these	DET
ap-1406	48	5	bases	basis	NOUN
ap-1406	48	6	beta	beta	NOUN
ap-1406	48	7	-	-	PUNCT
ap-1406	48	8	numbers	number	NOUN
ap-1406	48	9	;	;	PUNCT
ap-1406	48	10	now	now	ADV
ap-1406	48	11	these	these	DET
ap-1406	48	12	numbers	number	NOUN
ap-1406	48	13	are	be	AUX
ap-1406	48	14	commonly	commonly	ADV
ap-1406	48	15	called	call	VERB
ap-1406	48	16	parry	parry	NOUN
ap-1406	48	17	numbers	number	NOUN
ap-1406	48	18	.	.	PUNCT
ap-1406	49	1	we	we	PRON
ap-1406	49	2	can	can	AUX
ap-1406	49	3	demonstrate	demonstrate	VERB
ap-1406	49	4	the	the	DET
ap-1406	49	5	exceptional	exceptional	ADJ
ap-1406	49	6	properties	property	NOUN
ap-1406	49	7	of	of	ADP
ap-1406	49	8	parry	parry	NOUN
ap-1406	49	9	numbers	number	NOUN
ap-1406	49	10	on	on	ADP
ap-1406	49	11	two	two	NUM
ap-1406	49	12	facts	fact	NOUN
ap-1406	49	13	:	:	PUNCT
ap-1406	49	14	•	•	SCONJ
ap-1406	49	15	the	the	DET
ap-1406	49	16	subshift	subshift	NOUN
ap-1406	49	17	sβ	sβ	NOUN
ap-1406	49	18	is	be	AUX
ap-1406	49	19	sofic	sofic	ADJ
ap-1406	49	20	if	if	SCONJ
ap-1406	49	21	and	and	CCONJ
ap-1406	49	22	only	only	ADV
ap-1406	49	23	if	if	SCONJ
ap-1406	49	24	β	β	NOUN
ap-1406	49	25	is	be	AUX
ap-1406	49	26	a	a	DET
ap-1406	49	27	parry	parry	NOUN
ap-1406	49	28	number	number	NOUN
ap-1406	49	29	[	[	X
ap-1406	49	30	6	6	NUM
ap-1406	49	31	]	]	PUNCT
ap-1406	49	32	.	.	PUNCT
ap-1406	50	1	59	59	NUM
ap-1406	50	2	acta	acta	PROPN
ap-1406	50	3	polytechnica	polytechnica	PROPN
ap-1406	50	4	vol	vol	NOUN
ap-1406	50	5	.	.	PUNCT
ap-1406	51	1	51	51	NUM
ap-1406	51	2	no	no	INTJ
ap-1406	51	3	.	.	PUNCT
ap-1406	52	1	4/2011	4/2011	NUM
ap-1406	52	2	•	•	NOUN
ap-1406	52	3	distances	distance	NOUN
ap-1406	52	4	between	between	ADP
ap-1406	52	5	consecutive	consecutive	ADJ
ap-1406	52	6	β	β	NOUN
ap-1406	52	7	-	-	NOUN
ap-1406	52	8	integers	integer	NOUN
ap-1406	52	9	take	take	VERB
ap-1406	52	10	finitely	finitely	ADV
ap-1406	52	11	many	many	ADJ
ap-1406	52	12	values	value	NOUN
ap-1406	52	13	if	if	SCONJ
ap-1406	52	14	and	and	CCONJ
ap-1406	52	15	only	only	ADV
ap-1406	52	16	if	if	SCONJ
ap-1406	52	17	β	β	NOUN
ap-1406	52	18	is	be	AUX
ap-1406	52	19	a	a	DET
ap-1406	52	20	parry	parry	NOUN
ap-1406	52	21	number	number	NOUN
ap-1406	52	22	[	[	X
ap-1406	52	23	15	15	NUM
ap-1406	52	24	]	]	PUNCT
ap-1406	52	25	.	.	PUNCT
ap-1406	53	1	recently	recently	ADV
ap-1406	53	2	,	,	PUNCT
ap-1406	53	3	ito	ito	PROPN
ap-1406	53	4	and	and	CCONJ
ap-1406	53	5	sadahiro	sadahiro	PROPN
ap-1406	54	1	[	[	X
ap-1406	54	2	5	5	NUM
ap-1406	54	3	]	]	PUNCT
ap-1406	54	4	suggested	suggest	VERB
ap-1406	54	5	a	a	DET
ap-1406	54	6	study	study	NOUN
ap-1406	54	7	of	of	ADP
ap-1406	54	8	positional	positional	ADJ
ap-1406	54	9	numeration	numeration	NOUN
ap-1406	54	10	systems	system	NOUN
ap-1406	54	11	with	with	ADP
ap-1406	54	12	a	a	DET
ap-1406	54	13	negative	negative	ADJ
ap-1406	54	14	base	base	NOUN
ap-1406	54	15	−β	−β	NOUN
ap-1406	54	16	,	,	PUNCT
ap-1406	54	17	where	where	SCONJ
ap-1406	54	18	β	β	X
ap-1406	54	19	>	>	X
ap-1406	54	20	1	1	X
ap-1406	54	21	.	.	PUNCT
ap-1406	55	1	the	the	DET
ap-1406	55	2	representation	representation	NOUN
ap-1406	55	3	of	of	ADP
ap-1406	55	4	real	real	ADJ
ap-1406	55	5	numbers	number	NOUN
ap-1406	55	6	in	in	ADP
ap-1406	55	7	such	such	DET
ap-1406	55	8	a	a	DET
ap-1406	55	9	system	system	NOUN
ap-1406	55	10	is	be	AUX
ap-1406	55	11	defined	define	VERB
ap-1406	55	12	using	use	VERB
ap-1406	55	13	the	the	DET
ap-1406	55	14	transformation	transformation	NOUN
ap-1406	55	15	t	t	NOUN
ap-1406	55	16	:	:	PUNCT
ap-1406	56	1	[	[	X
ap-1406	56	2	lβ	lβ	ADP
ap-1406	56	3	,	,	PUNCT
ap-1406	56	4	rβ	rβ	X
ap-1406	56	5	)	)	PUNCT
ap-1406	56	6	�	�	PROPN
ap-1406	56	7	→	→	SYM
ap-1406	56	8	[	[	X
ap-1406	56	9	lβ	lβ	INTJ
ap-1406	56	10	,	,	PUNCT
ap-1406	56	11	rβ	rβ	PROPN
ap-1406	56	12	)	)	PUNCT
ap-1406	56	13	,	,	PUNCT
ap-1406	56	14	where	where	SCONJ
ap-1406	56	15	lβ	lβ	ADP
ap-1406	56	16	=	=	PUNCT
ap-1406	56	17	−	−	PROPN
ap-1406	56	18	β	β	X
ap-1406	56	19	β	β	NOUN
ap-1406	56	20	+	+	ADV
ap-1406	56	21	1	1	NUM
ap-1406	56	22	,	,	PUNCT
ap-1406	56	23	rβ	rβ	VERB
ap-1406	56	24	=	=	SYM
ap-1406	56	25	1	1	NUM
ap-1406	56	26	+	+	NUM
ap-1406	56	27	lβ	lβ	X
ap-1406	56	28	=	=	SYM
ap-1406	56	29	1	1	NUM
ap-1406	56	30	1	1	NUM
ap-1406	56	31	+	+	CCONJ
ap-1406	56	32	β	β	X
ap-1406	56	33	,	,	PUNCT
ap-1406	56	34	t	t	PROPN
ap-1406	56	35	(	(	PUNCT
ap-1406	56	36	x	x	NOUN
ap-1406	56	37	)	)	PUNCT
ap-1406	56	38	:	:	PUNCT
ap-1406	56	39	=	=	PUNCT
ap-1406	56	40	−βx	−βx	NOUN
ap-1406	56	41	−	−	PROPN
ap-1406	56	42	#	#	NOUN
ap-1406	56	43	−βx	−βx	NOUN
ap-1406	56	44	−	−	PROPN
ap-1406	56	45	lβ$	lβ$	PROPN
ap-1406	56	46	.	.	PUNCT
ap-1406	57	1	(	(	PUNCT
ap-1406	57	2	4	4	X
ap-1406	57	3	)	)	PUNCT
ap-1406	57	4	every	every	DET
ap-1406	57	5	real	real	ADJ
ap-1406	57	6	x	x	SYM
ap-1406	57	7	∈	∈	NOUN
ap-1406	57	8	iβ	iβ	ADP
ap-1406	57	9	:	:	PUNCT
ap-1406	57	10	=	=	SYM
ap-1406	58	1	[	[	X
ap-1406	58	2	lβ	lβ	ADP
ap-1406	58	3	,	,	PUNCT
ap-1406	58	4	rβ	rβ	NOUN
ap-1406	58	5	)	)	PUNCT
ap-1406	58	6	can	can	AUX
ap-1406	58	7	be	be	AUX
ap-1406	58	8	written	write	VERB
ap-1406	58	9	as	as	ADP
ap-1406	58	10	x	x	X
ap-1406	58	11	=	=	SYM
ap-1406	58	12	∞∑	∞∑	NOUN
ap-1406	58	13	i=1	i=1	PRON
ap-1406	58	14	xi	xi	X
ap-1406	58	15	(	(	PUNCT
ap-1406	58	16	−β)i	−β)i	NOUN
ap-1406	58	17	,	,	PUNCT
ap-1406	58	18	(	(	PUNCT
ap-1406	58	19	5	5	NUM
ap-1406	58	20	)	)	PUNCT
ap-1406	58	21	where	where	SCONJ
ap-1406	58	22	xi	xi	AUX
ap-1406	58	23	=	=	NOUN
ap-1406	58	24	#	#	SYM
ap-1406	58	25	−βt	−βt	PROPN
ap-1406	58	26	i−1(x	i−1(x	NOUN
ap-1406	58	27	)	)	PUNCT
ap-1406	58	28	−	−	ADP
ap-1406	58	29	lβ$	lβ$	PROPN
ap-1406	58	30	for	for	ADP
ap-1406	58	31	i	i	PRON
ap-1406	58	32	=	=	NOUN
ap-1406	58	33	1	1	NUM
ap-1406	58	34	,	,	PUNCT
ap-1406	58	35	2	2	NUM
ap-1406	58	36	,	,	PUNCT
ap-1406	58	37	3	3	NUM
ap-1406	58	38	,	,	PUNCT
ap-1406	58	39	.	.	PUNCT
ap-1406	58	40	.	.	PUNCT
ap-1406	58	41	.	.	PUNCT
ap-1406	59	1	the	the	DET
ap-1406	59	2	above	above	ADJ
ap-1406	59	3	expression	expression	NOUN
ap-1406	59	4	is	be	AUX
ap-1406	59	5	called	call	VERB
ap-1406	59	6	the	the	DET
ap-1406	59	7	(	(	PUNCT
ap-1406	59	8	−β)expansion	−β)expansion	NOUN
ap-1406	59	9	of	of	ADP
ap-1406	59	10	x.	x.	NOUN
ap-1406	59	11	it	it	PRON
ap-1406	59	12	can	can	AUX
ap-1406	59	13	also	also	ADV
ap-1406	59	14	be	be	AUX
ap-1406	59	15	written	write	VERB
ap-1406	59	16	as	as	ADP
ap-1406	59	17	the	the	DET
ap-1406	59	18	infinite	infinite	ADJ
ap-1406	59	19	word	word	NOUN
ap-1406	59	20	d−β(x	d−β(x	NOUN
ap-1406	59	21	)	)	PUNCT
ap-1406	59	22	=	=	SYM
ap-1406	59	23	x1x2x3	x1x2x3	PROPN
ap-1406	59	24	.	.	PUNCT
ap-1406	59	25	.	.	PUNCT
ap-1406	60	1	.	.	PUNCT
ap-1406	61	1	we	we	PRON
ap-1406	61	2	can	can	AUX
ap-1406	61	3	easily	easily	ADV
ap-1406	61	4	show	show	VERB
ap-1406	61	5	from	from	ADP
ap-1406	61	6	(	(	PUNCT
ap-1406	61	7	4	4	NUM
ap-1406	61	8	)	)	PUNCT
ap-1406	61	9	that	that	SCONJ
ap-1406	61	10	the	the	DET
ap-1406	61	11	digits	digit	NOUN
ap-1406	61	12	xi	xi	ADP
ap-1406	61	13	,	,	PUNCT
ap-1406	61	14	i	i	PRON
ap-1406	61	15	≥	≥	VERB
ap-1406	61	16	1	1	NUM
ap-1406	61	17	,	,	PUNCT
ap-1406	61	18	take	take	VERB
ap-1406	61	19	values	value	NOUN
ap-1406	61	20	in	in	ADP
ap-1406	61	21	the	the	DET
ap-1406	61	22	set	set	NOUN
ap-1406	61	23	a	a	X
ap-1406	61	24	=	=	PUNCT
ap-1406	61	25	{	{	PUNCT
ap-1406	61	26	0	0	NUM
ap-1406	61	27	,	,	PUNCT
ap-1406	61	28	1	1	NUM
ap-1406	61	29	,	,	PUNCT
ap-1406	61	30	2	2	NUM
ap-1406	61	31	,	,	PUNCT
ap-1406	61	32	.	.	PUNCT
ap-1406	61	33	.	.	PUNCT
ap-1406	61	34	.	.	PUNCT
ap-1406	62	1	,	,	PUNCT
ap-1406	62	2	#	#	SYM
ap-1406	62	3	β$	β$	NOUN
ap-1406	62	4	}	}	PUNCT
ap-1406	62	5	.	.	PUNCT
ap-1406	63	1	in	in	ADP
ap-1406	63	2	this	this	DET
ap-1406	63	3	case	case	NOUN
ap-1406	63	4	,	,	PUNCT
ap-1406	63	5	the	the	DET
ap-1406	63	6	ordering	ordering	NOUN
ap-1406	63	7	on	on	ADP
ap-1406	63	8	the	the	DET
ap-1406	63	9	set	set	NOUN
ap-1406	63	10	of	of	ADP
ap-1406	63	11	infinite	infinite	ADJ
ap-1406	63	12	words	word	NOUN
ap-1406	63	13	over	over	ADP
ap-1406	63	14	the	the	DET
ap-1406	63	15	alphabet	alphabet	NOUN
ap-1406	63	16	a	a	PRON
ap-1406	63	17	which	which	PRON
ap-1406	63	18	would	would	AUX
ap-1406	63	19	correspond	correspond	VERB
ap-1406	63	20	to	to	ADP
ap-1406	63	21	the	the	DET
ap-1406	63	22	ordering	ordering	NOUN
ap-1406	63	23	of	of	ADP
ap-1406	63	24	real	real	ADJ
ap-1406	63	25	numbers	number	NOUN
ap-1406	63	26	is	be	AUX
ap-1406	63	27	the	the	DET
ap-1406	63	28	so	so	ADV
ap-1406	63	29	-	-	PUNCT
ap-1406	63	30	called	call	VERB
ap-1406	63	31	alternate	alternate	ADJ
ap-1406	63	32	ordering	ordering	NOUN
ap-1406	63	33	:	:	PUNCT
ap-1406	63	34	we	we	PRON
ap-1406	63	35	say	say	VERB
ap-1406	63	36	that	that	SCONJ
ap-1406	63	37	x1x2x3	x1x2x3	PROPN
ap-1406	63	38	.	.	PUNCT
ap-1406	63	39	.	.	PUNCT
ap-1406	63	40	.	.	PUNCT
ap-1406	64	1	≺alt	≺alt	NOUN
ap-1406	64	2	y1y2y3	y1y2y3	NOUN
ap-1406	64	3	.	.	PUNCT
ap-1406	64	4	.	.	PUNCT
ap-1406	65	1	.	.	PUNCT
ap-1406	66	1	if	if	SCONJ
ap-1406	66	2	for	for	ADP
ap-1406	66	3	the	the	DET
ap-1406	66	4	minimal	minimal	ADJ
ap-1406	66	5	index	index	NOUN
ap-1406	66	6	j	j	PROPN
ap-1406	66	7	such	such	ADJ
ap-1406	66	8	that	that	SCONJ
ap-1406	66	9	xj	xj	PROPN
ap-1406	66	10	�	�	PROPN
ap-1406	66	11	=	=	SYM
ap-1406	66	12	yj	yj	PROPN
ap-1406	66	13	it	it	PRON
ap-1406	66	14	holds	hold	VERB
ap-1406	66	15	that	that	PRON
ap-1406	66	16	xj(−1)j	xj(−1)j	PROPN
ap-1406	66	17	<	<	X
ap-1406	66	18	yj(−1)j	yj(−1)j	PROPN
ap-1406	66	19	.	.	PUNCT
ap-1406	67	1	in	in	ADP
ap-1406	67	2	this	this	DET
ap-1406	67	3	notation	notation	NOUN
ap-1406	67	4	,	,	PUNCT
ap-1406	67	5	we	we	PRON
ap-1406	67	6	can	can	AUX
ap-1406	67	7	write	write	VERB
ap-1406	67	8	for	for	ADP
ap-1406	67	9	arbitrary	arbitrary	ADJ
ap-1406	67	10	x	x	NOUN
ap-1406	67	11	,	,	PUNCT
ap-1406	67	12	y	y	PROPN
ap-1406	67	13	∈	∈	PROPN
ap-1406	67	14	iβ	iβ	ADP
ap-1406	67	15	that	that	DET
ap-1406	67	16	x	x	X
ap-1406	67	17	≤	≤	ADJ
ap-1406	67	18	y	y	NUM
ap-1406	67	19	⇐	⇐	ADJ
ap-1406	67	20	⇒	⇒	PROPN
ap-1406	67	21	d−β(x	d−β(x	NOUN
ap-1406	67	22	)	)	PUNCT
ap-1406	67	23	)	)	PUNCT
ap-1406	67	24	alt	alt	VERB
ap-1406	67	25	d−β(y	d−β(y	NOUN
ap-1406	67	26	)	)	PUNCT
ap-1406	67	27	.	.	PUNCT
ap-1406	68	1	in	in	ADP
ap-1406	68	2	their	their	PRON
ap-1406	68	3	paper	paper	NOUN
ap-1406	68	4	,	,	PUNCT
ap-1406	68	5	ito	ito	PROPN
ap-1406	68	6	and	and	CCONJ
ap-1406	68	7	sadahiro	sadahiro	PROPN
ap-1406	68	8	have	have	AUX
ap-1406	68	9	provided	provide	VERB
ap-1406	68	10	a	a	DET
ap-1406	68	11	criterion	criterion	NOUN
ap-1406	68	12	to	to	PART
ap-1406	68	13	decide	decide	VERB
ap-1406	68	14	whether	whether	SCONJ
ap-1406	68	15	an	an	DET
ap-1406	68	16	infinite	infinite	ADJ
ap-1406	68	17	word	word	NOUN
ap-1406	68	18	an	an	DET
ap-1406	68	19	belongs	belong	VERB
ap-1406	68	20	to	to	ADP
ap-1406	68	21	the	the	DET
ap-1406	68	22	set	set	NOUN
ap-1406	68	23	of	of	ADP
ap-1406	68	24	(	(	PUNCT
ap-1406	68	25	−β)-expansions	−β)-expansion	NOUN
ap-1406	68	26	,	,	PUNCT
ap-1406	68	27	i.e.	i.e.	X
ap-1406	68	28	to	to	ADP
ap-1406	68	29	the	the	DET
ap-1406	68	30	set	set	NOUN
ap-1406	68	31	d−β	d−β	NOUN
ap-1406	68	32	=	=	SYM
ap-1406	68	33	{	{	PUNCT
ap-1406	68	34	d−β(x	d−β(x	NOUN
ap-1406	68	35	)	)	PUNCT
ap-1406	68	36	|	|	ADV
ap-1406	68	37	x	x	SYM
ap-1406	68	38	∈	∈	PROPN
ap-1406	68	39	iβ	iβ	ADP
ap-1406	68	40	}	}	PUNCT
ap-1406	68	41	.	.	PUNCT
ap-1406	69	1	this	this	DET
ap-1406	69	2	time	time	NOUN
ap-1406	69	3	,	,	PUNCT
ap-1406	69	4	the	the	DET
ap-1406	69	5	criterion	criterion	NOUN
ap-1406	69	6	is	be	AUX
ap-1406	69	7	given	give	VERB
ap-1406	69	8	in	in	ADP
ap-1406	69	9	terms	term	NOUN
ap-1406	69	10	of	of	ADP
ap-1406	69	11	two	two	NUM
ap-1406	69	12	infinite	infinite	ADJ
ap-1406	69	13	words	word	NOUN
ap-1406	69	14	,	,	PUNCT
ap-1406	69	15	namely	namely	ADV
ap-1406	69	16	d−β(lβ	d−β(lβ	NOUN
ap-1406	69	17	)	)	PUNCT
ap-1406	69	18	and	and	CCONJ
ap-1406	69	19	d∗−β(rβ	d∗−β(rβ	NOUN
ap-1406	69	20	)	)	PUNCT
ap-1406	69	21	:	:	PUNCT
ap-1406	70	1	=	=	PUNCT
ap-1406	70	2	lim	lim	PROPN
ap-1406	70	3	ε→0	ε→0	NOUN
ap-1406	70	4	+	+	CCONJ
ap-1406	70	5	d−β(rβ	d−β(rβ	PROPN
ap-1406	70	6	−	−	PROPN
ap-1406	70	7	ε	ε	PROPN
ap-1406	70	8	)	)	PUNCT
ap-1406	70	9	.	.	PUNCT
ap-1406	71	1	these	these	DET
ap-1406	71	2	two	two	NUM
ap-1406	71	3	infinite	infinite	ADJ
ap-1406	71	4	words	word	NOUN
ap-1406	71	5	have	have	VERB
ap-1406	71	6	a	a	DET
ap-1406	71	7	close	close	ADJ
ap-1406	71	8	relation	relation	NOUN
ap-1406	71	9	:	:	PUNCT
ap-1406	71	10	if	if	SCONJ
ap-1406	71	11	d−β(lβ	d−β(lβ	NOUN
ap-1406	71	12	)	)	PUNCT
ap-1406	71	13	is	be	AUX
ap-1406	71	14	purely	purely	ADV
ap-1406	71	15	periodic	periodic	ADJ
ap-1406	71	16	with	with	ADP
ap-1406	71	17	odd	odd	ADJ
ap-1406	71	18	period	period	NOUN
ap-1406	71	19	length	length	NOUN
ap-1406	71	20	,	,	PUNCT
ap-1406	71	21	i.e.	i.e.	X
ap-1406	71	22	d−β(lβ	d−β(lβ	NOUN
ap-1406	71	23	)	)	PUNCT
ap-1406	71	24	=	=	PUNCT
ap-1406	72	1	(	(	PUNCT
ap-1406	72	2	d1d2	d1d2	X
ap-1406	72	3	.	.	PUNCT
ap-1406	72	4	.	.	PUNCT
ap-1406	72	5	.	.	PUNCT
ap-1406	73	1	d2k+1)ω	d2k+1)ω	PROPN
ap-1406	73	2	,	,	PUNCT
ap-1406	73	3	then	then	ADV
ap-1406	73	4	we	we	PRON
ap-1406	73	5	have	have	VERB
ap-1406	73	6	d∗−β(rβ	d∗−β(rβ	NOUN
ap-1406	73	7	)	)	PUNCT
ap-1406	73	8	=(	=(	NOUN
ap-1406	73	9	0d1d2	0d1d2	NUM
ap-1406	73	10	.	.	PUNCT
ap-1406	73	11	.	.	PUNCT
ap-1406	73	12	.	.	PUNCT
ap-1406	74	1	(	(	PUNCT
ap-1406	74	2	d2k+1	d2k+1	VERB
ap-1406	74	3	−	−	NOUN
ap-1406	74	4	1	1	NUM
ap-1406	74	5	)	)	PUNCT
ap-1406	74	6	)	)	PUNCT
ap-1406	75	1	ω	ω	X
ap-1406	75	2	.	.	PUNCT
ap-1406	76	1	(	(	PUNCT
ap-1406	76	2	as	as	ADP
ap-1406	76	3	usual	usual	ADJ
ap-1406	76	4	,	,	PUNCT
ap-1406	76	5	the	the	DET
ap-1406	76	6	notation	notation	NOUN
ap-1406	76	7	wω	wω	NOUN
ap-1406	76	8	stands	stand	VERB
ap-1406	76	9	for	for	ADP
ap-1406	76	10	infinite	infinite	ADJ
ap-1406	76	11	repetition	repetition	NOUN
ap-1406	76	12	of	of	ADP
ap-1406	76	13	the	the	DET
ap-1406	76	14	string	string	NOUN
ap-1406	76	15	w.	w.	PROPN
ap-1406	76	16	)	)	PUNCT
ap-1406	76	17	in	in	ADP
ap-1406	76	18	all	all	DET
ap-1406	76	19	other	other	ADJ
ap-1406	76	20	cases	case	NOUN
ap-1406	76	21	we	we	PRON
ap-1406	76	22	have	have	VERB
ap-1406	76	23	d∗−β(rβ	d∗−β(rβ	NOUN
ap-1406	76	24	)	)	PUNCT
ap-1406	77	1	=	=	SYM
ap-1406	77	2	0d−β(lβ	0d−β(lβ	NOUN
ap-1406	77	3	)	)	PUNCT
ap-1406	77	4	.	.	PUNCT
ap-1406	78	1	ito	ito	PROPN
ap-1406	78	2	and	and	CCONJ
ap-1406	78	3	sadahiro	sadahiro	PROPN
ap-1406	78	4	have	have	AUX
ap-1406	78	5	shown	show	VERB
ap-1406	78	6	that	that	SCONJ
ap-1406	78	7	an	an	DET
ap-1406	78	8	infinite	infinite	ADJ
ap-1406	78	9	word	word	NOUN
ap-1406	78	10	x1x2x3	x1x2x3	PROPN
ap-1406	78	11	.	.	PUNCT
ap-1406	78	12	.	.	PUNCT
ap-1406	78	13	.	.	PUNCT
ap-1406	79	1	represents	represent	VERB
ap-1406	79	2	a	a	DET
ap-1406	79	3	(	(	PUNCT
ap-1406	79	4	−β)-expansion	−β)-expansion	NOUN
ap-1406	79	5	of	of	ADP
ap-1406	79	6	some	some	DET
ap-1406	79	7	x	x	SYM
ap-1406	79	8	∈	∈	PROPN
ap-1406	80	1	[	[	X
ap-1406	80	2	lβ	lβ	INTJ
ap-1406	80	3	,	,	PUNCT
ap-1406	80	4	rβ	rβ	X
ap-1406	80	5	)	)	PUNCT
ap-1406	80	6	if	if	SCONJ
ap-1406	80	7	and	and	CCONJ
ap-1406	80	8	only	only	ADV
ap-1406	80	9	if	if	SCONJ
ap-1406	80	10	for	for	ADP
ap-1406	80	11	every	every	DET
ap-1406	80	12	i	i	PRON
ap-1406	80	13	≥	≥	NOUN
ap-1406	80	14	1	1	NUM
ap-1406	80	15	it	it	PRON
ap-1406	80	16	holds	hold	VERB
ap-1406	80	17	that	that	DET
ap-1406	80	18	d−β(lβ	d−β(lβ	NOUN
ap-1406	80	19	)	)	PUNCT
ap-1406	80	20	)	)	PUNCT
ap-1406	80	21	alt	alt	VERB
ap-1406	80	22	xixi+1xi+2	xixi+1xi+2	PROPN
ap-1406	80	23	.	.	PUNCT
ap-1406	80	24	.	.	PUNCT
ap-1406	80	25	.	.	PUNCT
ap-1406	81	1	≺alt	≺alt	NOUN
ap-1406	81	2	d∗−β(rβ	d∗−β(rβ	NOUN
ap-1406	81	3	)	)	PUNCT
ap-1406	81	4	.	.	PUNCT
ap-1406	82	1	(	(	PUNCT
ap-1406	82	2	6	6	X
ap-1406	82	3	)	)	PUNCT
ap-1406	82	4	the	the	DET
ap-1406	82	5	above	above	ADJ
ap-1406	82	6	condition	condition	NOUN
ap-1406	82	7	ensures	ensure	VERB
ap-1406	82	8	that	that	SCONJ
ap-1406	82	9	the	the	DET
ap-1406	82	10	set	set	ADJ
ap-1406	82	11	d−β	d−β	NOUN
ap-1406	82	12	of	of	ADP
ap-1406	82	13	infinite	infinite	ADJ
ap-1406	82	14	words	word	NOUN
ap-1406	82	15	representing	represent	VERB
ap-1406	82	16	(	(	PUNCT
ap-1406	82	17	−β)-expansions	−β)-expansion	NOUN
ap-1406	82	18	is	be	AUX
ap-1406	82	19	shift	shift	NOUN
ap-1406	82	20	invariant	invariant	ADJ
ap-1406	82	21	.	.	PUNCT
ap-1406	83	1	in	in	ADP
ap-1406	83	2	[	[	X
ap-1406	83	3	5	5	X
ap-1406	83	4	]	]	PUNCT
ap-1406	83	5	it	it	PRON
ap-1406	83	6	is	be	AUX
ap-1406	83	7	shown	show	VERB
ap-1406	83	8	that	that	SCONJ
ap-1406	83	9	the	the	DET
ap-1406	83	10	closure	closure	NOUN
ap-1406	83	11	of	of	ADP
ap-1406	83	12	d−β	d−β	NOUN
ap-1406	83	13	defines	define	VERB
ap-1406	83	14	a	a	DET
ap-1406	83	15	sofic	sofic	ADJ
ap-1406	83	16	system	system	NOUN
ap-1406	83	17	if	if	SCONJ
ap-1406	83	18	and	and	CCONJ
ap-1406	83	19	only	only	ADV
ap-1406	83	20	if	if	SCONJ
ap-1406	83	21	d−β(lβ	d−β(lβ	NOUN
ap-1406	83	22	)	)	PUNCT
ap-1406	83	23	is	be	AUX
ap-1406	83	24	eventually	eventually	ADV
ap-1406	83	25	periodic	periodic	ADJ
ap-1406	83	26	.	.	PUNCT
ap-1406	84	1	by	by	ADP
ap-1406	84	2	analogy	analogy	NOUN
ap-1406	84	3	with	with	ADP
ap-1406	84	4	the	the	DET
ap-1406	84	5	definition	definition	NOUN
ap-1406	84	6	of	of	ADP
ap-1406	84	7	parry	parry	PROPN
ap-1406	84	8	numbers	number	NOUN
ap-1406	84	9	,	,	PUNCT
ap-1406	84	10	we	we	PRON
ap-1406	84	11	suggest	suggest	VERB
ap-1406	84	12	that	that	SCONJ
ap-1406	84	13	numbers	number	NOUN
ap-1406	84	14	β	β	X
ap-1406	84	15	>	>	X
ap-1406	84	16	1	1	NUM
ap-1406	84	17	such	such	ADJ
ap-1406	84	18	that	that	DET
ap-1406	84	19	d−β(lβ	d−β(lβ	NOUN
ap-1406	84	20	)	)	PUNCT
ap-1406	84	21	is	be	AUX
ap-1406	84	22	eventually	eventually	ADV
ap-1406	84	23	periodic	periodic	ADJ
ap-1406	84	24	be	be	AUX
ap-1406	84	25	called	call	VERB
ap-1406	84	26	ito	ito	PROPN
ap-1406	84	27	-	-	PUNCT
ap-1406	84	28	sadahiro	sadahiro	NOUN
ap-1406	84	29	numbers	number	NOUN
ap-1406	84	30	.	.	PUNCT
ap-1406	85	1	the	the	DET
ap-1406	85	2	relation	relation	NOUN
ap-1406	85	3	of	of	ADP
ap-1406	85	4	the	the	DET
ap-1406	85	5	set	set	NOUN
ap-1406	85	6	of	of	ADP
ap-1406	85	7	ito	ito	PROPN
ap-1406	85	8	-	-	PUNCT
ap-1406	85	9	sadahiro	sadahiro	PROPN
ap-1406	85	10	numbers	number	NOUN
ap-1406	85	11	and	and	CCONJ
ap-1406	85	12	the	the	DET
ap-1406	85	13	set	set	NOUN
ap-1406	85	14	of	of	ADP
ap-1406	85	15	parry	parry	PROPN
ap-1406	85	16	numbers	number	NOUN
ap-1406	85	17	is	be	AUX
ap-1406	85	18	not	not	PART
ap-1406	85	19	obvious	obvious	ADJ
ap-1406	85	20	.	.	PUNCT
ap-1406	86	1	bassino	bassino	NOUN
ap-1406	87	1	[	[	X
ap-1406	87	2	2	2	X
ap-1406	87	3	]	]	PUNCT
ap-1406	87	4	has	have	AUX
ap-1406	87	5	shown	show	VERB
ap-1406	87	6	that	that	SCONJ
ap-1406	87	7	quadratic	quadratic	ADJ
ap-1406	87	8	numbers	number	NOUN
ap-1406	87	9	,	,	PUNCT
ap-1406	87	10	as	as	ADV
ap-1406	87	11	well	well	ADV
ap-1406	87	12	as	as	ADP
ap-1406	87	13	cubic	cubic	ADJ
ap-1406	87	14	numbers	number	NOUN
ap-1406	87	15	which	which	PRON
ap-1406	87	16	are	be	AUX
ap-1406	87	17	not	not	PART
ap-1406	87	18	totally	totally	ADV
ap-1406	87	19	real	real	ADJ
ap-1406	87	20	,	,	PUNCT
ap-1406	87	21	are	be	AUX
ap-1406	87	22	parry	parry	ADJ
ap-1406	87	23	if	if	SCONJ
ap-1406	87	24	and	and	CCONJ
ap-1406	87	25	only	only	ADV
ap-1406	87	26	if	if	SCONJ
ap-1406	87	27	they	they	PRON
ap-1406	87	28	are	be	AUX
ap-1406	87	29	pisot	pisot	ADJ
ap-1406	87	30	.	.	PUNCT
ap-1406	88	1	for	for	ADP
ap-1406	88	2	the	the	DET
ap-1406	88	3	same	same	ADJ
ap-1406	88	4	class	class	NOUN
ap-1406	88	5	of	of	ADP
ap-1406	88	6	numbers	number	NOUN
ap-1406	88	7	,	,	PUNCT
ap-1406	88	8	we	we	PRON
ap-1406	88	9	prove	prove	VERB
ap-1406	88	10	in	in	ADP
ap-1406	88	11	[	[	X
ap-1406	88	12	10	10	NUM
ap-1406	88	13	]	]	PUNCT
ap-1406	88	14	that	that	SCONJ
ap-1406	88	15	β	β	PROPN
ap-1406	88	16	is	be	AUX
ap-1406	88	17	ito	ito	PROPN
ap-1406	88	18	-	-	NOUN
ap-1406	88	19	sadahiro	sadahiro	NOUN
ap-1406	88	20	if	if	SCONJ
ap-1406	88	21	and	and	CCONJ
ap-1406	88	22	only	only	ADV
ap-1406	88	23	if	if	SCONJ
ap-1406	88	24	it	it	PRON
ap-1406	88	25	is	be	AUX
ap-1406	88	26	pisot	pisot	ADJ
ap-1406	88	27	.	.	PUNCT
ap-1406	89	1	this	this	PRON
ap-1406	89	2	means	mean	VERB
ap-1406	89	3	that	that	SCONJ
ap-1406	89	4	notions	notion	NOUN
ap-1406	89	5	of	of	ADP
ap-1406	89	6	parry	parry	NOUN
ap-1406	89	7	numbers	number	NOUN
ap-1406	89	8	and	and	CCONJ
ap-1406	89	9	ito	ito	PROPN
ap-1406	89	10	-	-	PROPN
ap-1406	89	11	sadahiro	sadahiro	PROPN
ap-1406	89	12	numbers	number	NOUN
ap-1406	89	13	on	on	ADP
ap-1406	89	14	the	the	DET
ap-1406	89	15	mentioned	mention	VERB
ap-1406	89	16	type	type	NOUN
ap-1406	89	17	of	of	ADP
ap-1406	89	18	irrationals	irrational	NOUN
ap-1406	89	19	do	do	AUX
ap-1406	89	20	not	not	PART
ap-1406	89	21	differ	differ	VERB
ap-1406	89	22	.	.	PUNCT
ap-1406	90	1	this	this	PRON
ap-1406	90	2	would	would	AUX
ap-1406	90	3	support	support	VERB
ap-1406	90	4	the	the	DET
ap-1406	90	5	hypothesis	hypothesis	NOUN
ap-1406	90	6	stated	state	VERB
ap-1406	90	7	in	in	ADP
ap-1406	90	8	the	the	DET
ap-1406	90	9	first	first	ADJ
ap-1406	90	10	version	version	NOUN
ap-1406	90	11	of	of	ADP
ap-1406	90	12	this	this	DET
ap-1406	90	13	paper	paper	NOUN
ap-1406	90	14	,	,	PUNCT
ap-1406	90	15	namely	namely	ADV
ap-1406	90	16	that	that	SCONJ
ap-1406	90	17	the	the	DET
ap-1406	90	18	set	set	NOUN
ap-1406	90	19	of	of	ADP
ap-1406	90	20	parry	parry	PROPN
ap-1406	90	21	numbers	number	NOUN
ap-1406	90	22	and	and	CCONJ
ap-1406	90	23	the	the	DET
ap-1406	90	24	set	set	NOUN
ap-1406	90	25	of	of	ADP
ap-1406	90	26	ito	ito	PROPN
ap-1406	90	27	-	-	PUNCT
ap-1406	90	28	sadahiro	sadahiro	PROPN
ap-1406	90	29	numbers	number	NOUN
ap-1406	90	30	coincide	coincide	VERB
ap-1406	90	31	.	.	PUNCT
ap-1406	91	1	however	however	ADV
ap-1406	91	2	,	,	PUNCT
ap-1406	91	3	during	during	ADP
ap-1406	91	4	the	the	DET
ap-1406	91	5	refereeing	referee	VERB
ap-1406	91	6	process	process	NOUN
ap-1406	91	7	liao	liao	NOUN
ap-1406	91	8	and	and	CCONJ
ap-1406	91	9	steiner	steiner	NOUN
ap-1406	91	10	[	[	X
ap-1406	91	11	9	9	NUM
ap-1406	91	12	]	]	PUNCT
ap-1406	91	13	found	find	VERB
ap-1406	91	14	an	an	DET
ap-1406	91	15	example	example	NOUN
ap-1406	91	16	of	of	ADP
ap-1406	91	17	a	a	DET
ap-1406	91	18	parry	parry	NOUN
ap-1406	91	19	number	number	NOUN
ap-1406	91	20	which	which	PRON
ap-1406	91	21	is	be	AUX
ap-1406	91	22	not	not	PART
ap-1406	91	23	an	an	DET
ap-1406	91	24	itosadahiro	itosadahiro	NOUN
ap-1406	91	25	number	number	NOUN
ap-1406	91	26	,	,	PUNCT
ap-1406	91	27	and	and	CCONJ
ap-1406	91	28	vice	vice	NOUN
ap-1406	91	29	-	-	NOUN
ap-1406	91	30	versa	versa	NOUN
ap-1406	91	31	.	.	PUNCT
ap-1406	92	1	the	the	DET
ap-1406	92	2	main	main	ADJ
ap-1406	92	3	results	result	NOUN
ap-1406	92	4	of	of	ADP
ap-1406	92	5	this	this	DET
ap-1406	92	6	paper	paper	NOUN
ap-1406	92	7	are	be	AUX
ap-1406	92	8	formulated	formulate	VERB
ap-1406	92	9	as	as	ADP
ap-1406	92	10	theorems	theorem	NOUN
ap-1406	92	11	4	4	NUM
ap-1406	92	12	and	and	CCONJ
ap-1406	92	13	7	7	NUM
ap-1406	92	14	.	.	X
ap-1406	92	15	theorem	theorem	NOUN
ap-1406	92	16	4	4	NUM
ap-1406	92	17	gives	give	VERB
ap-1406	92	18	a	a	DET
ap-1406	92	19	bound	bind	VERB
ap-1406	92	20	on	on	ADP
ap-1406	92	21	the	the	DET
ap-1406	92	22	modulus	modulus	NOUN
ap-1406	92	23	of	of	ADP
ap-1406	92	24	conjugates	conjugate	NOUN
ap-1406	92	25	of	of	ADP
ap-1406	92	26	ito	ito	PROPN
ap-1406	92	27	-	-	PUNCT
ap-1406	92	28	sadahiro	sadahiro	PROPN
ap-1406	92	29	numbers	number	NOUN
ap-1406	92	30	;	;	PUNCT
ap-1406	92	31	theorem	theorem	VERB
ap-1406	92	32	7	7	NUM
ap-1406	92	33	shows	show	VERB
ap-1406	92	34	that	that	SCONJ
ap-1406	92	35	periodicity	periodicity	NOUN
ap-1406	92	36	of	of	ADP
ap-1406	92	37	(	(	PUNCT
ap-1406	92	38	−β)-expansion	−β)-expansion	NOUN
ap-1406	92	39	of	of	ADP
ap-1406	92	40	all	all	DET
ap-1406	92	41	numbers	number	NOUN
ap-1406	92	42	in	in	ADP
ap-1406	92	43	the	the	DET
ap-1406	92	44	field	field	NOUN
ap-1406	92	45	q(β	q(β	NOUN
ap-1406	92	46	)	)	PUNCT
ap-1406	92	47	requires	require	VERB
ap-1406	92	48	β	β	NOUN
ap-1406	92	49	to	to	PART
ap-1406	92	50	be	be	AUX
ap-1406	92	51	a	a	DET
ap-1406	92	52	pisot	pisot	NOUN
ap-1406	92	53	or	or	CCONJ
ap-1406	92	54	salem	salem	NOUN
ap-1406	92	55	number	number	NOUN
ap-1406	92	56	.	.	PUNCT
ap-1406	93	1	statements	statement	NOUN
ap-1406	93	2	which	which	PRON
ap-1406	93	3	we	we	PRON
ap-1406	93	4	prove	prove	VERB
ap-1406	93	5	,	,	PUNCT
ap-1406	93	6	as	as	ADV
ap-1406	93	7	well	well	ADV
ap-1406	93	8	as	as	ADP
ap-1406	93	9	results	result	NOUN
ap-1406	93	10	of	of	ADP
ap-1406	93	11	other	other	ADJ
ap-1406	93	12	authors	author	NOUN
ap-1406	93	13	that	that	PRON
ap-1406	93	14	we	we	PRON
ap-1406	93	15	recall	recall	VERB
ap-1406	93	16	,	,	PUNCT
ap-1406	93	17	demonstrate	demonstrate	VERB
ap-1406	93	18	similarities	similarity	NOUN
ap-1406	93	19	between	between	ADP
ap-1406	93	20	the	the	DET
ap-1406	93	21	behaviour	behaviour	NOUN
ap-1406	93	22	of	of	ADP
ap-1406	93	23	β	β	NOUN
ap-1406	93	24	-	-	NOUN
ap-1406	93	25	expansions	expansion	NOUN
ap-1406	93	26	and	and	CCONJ
ap-1406	93	27	(	(	PUNCT
ap-1406	93	28	−β)-expansions	−β)-expansion	NOUN
ap-1406	93	29	.	.	PUNCT
ap-1406	94	1	we	we	PRON
ap-1406	94	2	mention	mention	VERB
ap-1406	94	3	also	also	ADV
ap-1406	94	4	phenomena	phenomenon	NOUN
ap-1406	94	5	in	in	ADP
ap-1406	94	6	which	which	PRON
ap-1406	94	7	the	the	DET
ap-1406	94	8	two	two	NUM
ap-1406	94	9	essentially	essentially	ADV
ap-1406	94	10	differ	differ	VERB
ap-1406	94	11	.	.	PUNCT
ap-1406	95	1	2	2	NUM
ap-1406	95	2	preliminaries	preliminary	NOUN
ap-1406	95	3	let	let	VERB
ap-1406	95	4	us	we	PRON
ap-1406	95	5	first	first	ADV
ap-1406	95	6	recall	recall	VERB
ap-1406	95	7	some	some	DET
ap-1406	95	8	number	number	NOUN
ap-1406	95	9	theoretical	theoretical	ADJ
ap-1406	95	10	notions	notion	NOUN
ap-1406	95	11	.	.	PUNCT
ap-1406	96	1	a	a	DET
ap-1406	96	2	complex	complex	ADJ
ap-1406	96	3	number	number	NOUN
ap-1406	96	4	β	β	NOUN
ap-1406	96	5	is	be	AUX
ap-1406	96	6	called	call	VERB
ap-1406	96	7	an	an	DET
ap-1406	96	8	algebraic	algebraic	ADJ
ap-1406	96	9	number	number	NOUN
ap-1406	96	10	,	,	PUNCT
ap-1406	96	11	if	if	SCONJ
ap-1406	96	12	it	it	PRON
ap-1406	96	13	is	be	AUX
ap-1406	96	14	a	a	DET
ap-1406	96	15	root	root	NOUN
ap-1406	96	16	of	of	ADP
ap-1406	96	17	a	a	DET
ap-1406	96	18	monic	monic	ADJ
ap-1406	96	19	polynomial	polynomial	NOUN
ap-1406	96	20	xn	xn	PUNCT
ap-1406	97	1	+	+	PROPN
ap-1406	97	2	an−1x	an−1x	PROPN
ap-1406	97	3	n−1	n−1	PROPN
ap-1406	97	4	+	+	PROPN
ap-1406	97	5	.	.	PUNCT
ap-1406	97	6	.	.	PUNCT
ap-1406	98	1	.+	.+	NOUN
ap-1406	98	2	a1x	a1x	PART
ap-1406	98	3	+	+	NUM
ap-1406	98	4	a0	a0	PROPN
ap-1406	98	5	,	,	PUNCT
ap-1406	98	6	with	with	ADP
ap-1406	98	7	rational	rational	ADJ
ap-1406	98	8	coefficients	coefficient	NOUN
ap-1406	98	9	a0	a0	PROPN
ap-1406	98	10	,	,	PUNCT
ap-1406	98	11	.	.	PUNCT
ap-1406	98	12	.	.	PUNCT
ap-1406	99	1	.	.	PUNCT
ap-1406	100	1	,	,	PUNCT
ap-1406	100	2	an−1	an−1	PROPN
ap-1406	100	3	∈	∈	PROPN
ap-1406	100	4	q.	q.	NOUN
ap-1406	100	5	a	a	DET
ap-1406	100	6	monic	monic	ADJ
ap-1406	100	7	polynomial	polynomial	NOUN
ap-1406	100	8	with	with	ADP
ap-1406	100	9	rational	rational	ADJ
ap-1406	100	10	coefficients	coefficient	NOUN
ap-1406	100	11	and	and	CCONJ
ap-1406	100	12	root	root	NOUN
ap-1406	100	13	β	β	X
ap-1406	100	14	of	of	ADP
ap-1406	100	15	the	the	DET
ap-1406	100	16	minimal	minimal	ADJ
ap-1406	100	17	degree	degree	NOUN
ap-1406	100	18	among	among	ADP
ap-1406	100	19	all	all	DET
ap-1406	100	20	polynomials	polynomial	NOUN
ap-1406	100	21	with	with	ADP
ap-1406	100	22	the	the	DET
ap-1406	100	23	same	same	ADJ
ap-1406	100	24	properties	property	NOUN
ap-1406	100	25	is	be	AUX
ap-1406	100	26	called	call	VERB
ap-1406	100	27	the	the	DET
ap-1406	100	28	minimal	minimal	ADJ
ap-1406	100	29	polynomial	polynomial	NOUN
ap-1406	100	30	of	of	ADP
ap-1406	100	31	β	β	NOUN
ap-1406	100	32	,	,	PUNCT
ap-1406	100	33	and	and	CCONJ
ap-1406	100	34	its	its	PRON
ap-1406	100	35	degree	degree	NOUN
ap-1406	100	36	is	be	AUX
ap-1406	100	37	called	call	VERB
ap-1406	100	38	the	the	DET
ap-1406	100	39	degree	degree	NOUN
ap-1406	100	40	of	of	ADP
ap-1406	100	41	β	β	NOUN
ap-1406	100	42	.	.	PUNCT
ap-1406	101	1	the	the	DET
ap-1406	101	2	roots	root	NOUN
ap-1406	101	3	of	of	ADP
ap-1406	101	4	the	the	DET
ap-1406	101	5	minimal	minimal	ADJ
ap-1406	101	6	polynomial	polynomial	NOUN
ap-1406	101	7	are	be	AUX
ap-1406	101	8	algebraic	algebraic	ADJ
ap-1406	101	9	conjugates	conjugate	NOUN
ap-1406	101	10	.	.	PUNCT
ap-1406	102	1	if	if	SCONJ
ap-1406	102	2	the	the	DET
ap-1406	102	3	minimal	minimal	ADJ
ap-1406	102	4	polynomial	polynomial	NOUN
ap-1406	102	5	of	of	ADP
ap-1406	102	6	β	β	PROPN
ap-1406	102	7	has	have	VERB
ap-1406	102	8	integer	integer	NOUN
ap-1406	102	9	coefficients	coefficient	NOUN
ap-1406	102	10	,	,	PUNCT
ap-1406	102	11	β	β	X
ap-1406	102	12	is	be	AUX
ap-1406	102	13	called	call	VERB
ap-1406	102	14	an	an	DET
ap-1406	102	15	algebraic	algebraic	ADJ
ap-1406	102	16	integer	integer	NOUN
ap-1406	102	17	.	.	PUNCT
ap-1406	103	1	an	an	DET
ap-1406	103	2	algebraic	algebraic	ADJ
ap-1406	103	3	integer	integer	NOUN
ap-1406	103	4	β	β	X
ap-1406	103	5	>	>	X
ap-1406	103	6	1	1	NUM
ap-1406	103	7	is	be	AUX
ap-1406	103	8	called	call	VERB
ap-1406	103	9	a	a	DET
ap-1406	103	10	perron	perron	PROPN
ap-1406	103	11	number	number	NOUN
ap-1406	103	12	,	,	PUNCT
ap-1406	103	13	if	if	SCONJ
ap-1406	103	14	all	all	DET
ap-1406	103	15	its	its	PRON
ap-1406	103	16	conjugates	conjugate	NOUN
ap-1406	103	17	are	be	AUX
ap-1406	103	18	in	in	ADP
ap-1406	103	19	modulus	modulus	NOUN
ap-1406	103	20	strictly	strictly	ADV
ap-1406	103	21	smaller	small	ADJ
ap-1406	103	22	than	than	ADP
ap-1406	103	23	β	β	X
ap-1406	103	24	.	.	PUNCT
ap-1406	104	1	an	an	DET
ap-1406	104	2	algebraic	algebraic	ADJ
ap-1406	104	3	integer	integer	NOUN
ap-1406	104	4	β	β	X
ap-1406	104	5	>	>	X
ap-1406	104	6	1	1	NUM
ap-1406	104	7	is	be	AUX
ap-1406	104	8	called	call	VERB
ap-1406	104	9	a	a	DET
ap-1406	104	10	pisot	pisot	ADJ
ap-1406	104	11	number	number	NOUN
ap-1406	104	12	,	,	PUNCT
ap-1406	104	13	if	if	SCONJ
ap-1406	104	14	all	all	DET
ap-1406	104	15	its	its	PRON
ap-1406	104	16	conjugates	conjugate	NOUN
ap-1406	104	17	are	be	AUX
ap-1406	104	18	in	in	ADP
ap-1406	104	19	modulus	modulus	NOUN
ap-1406	104	20	strictly	strictly	ADV
ap-1406	104	21	smaller	small	ADJ
ap-1406	104	22	than	than	ADP
ap-1406	104	23	1	1	NUM
ap-1406	104	24	.	.	PUNCT
ap-1406	105	1	an	an	DET
ap-1406	105	2	algebraic	algebraic	ADJ
ap-1406	105	3	integer	integer	NOUN
ap-1406	105	4	β	β	X
ap-1406	105	5	>	>	X
ap-1406	105	6	1	1	NUM
ap-1406	105	7	is	be	AUX
ap-1406	105	8	called	call	VERB
ap-1406	105	9	a	a	DET
ap-1406	105	10	salem	salem	NOUN
ap-1406	105	11	number	number	NOUN
ap-1406	105	12	,	,	PUNCT
ap-1406	105	13	if	if	SCONJ
ap-1406	105	14	all	all	DET
ap-1406	105	15	its	its	PRON
ap-1406	105	16	conjugates	conjugate	NOUN
ap-1406	105	17	are	be	AUX
ap-1406	105	18	in	in	ADP
ap-1406	105	19	modulus	modulus	NOUN
ap-1406	105	20	smaller	small	ADJ
ap-1406	105	21	than	than	ADP
ap-1406	105	22	or	or	CCONJ
ap-1406	105	23	equal	equal	ADJ
ap-1406	105	24	to	to	ADP
ap-1406	105	25	1	1	NUM
ap-1406	105	26	and	and	CCONJ
ap-1406	105	27	β	β	X
ap-1406	105	28	is	be	AUX
ap-1406	105	29	not	not	PART
ap-1406	105	30	a	a	DET
ap-1406	105	31	pisot	pisot	ADJ
ap-1406	105	32	number	number	NOUN
ap-1406	105	33	.	.	PUNCT
ap-1406	106	1	if	if	SCONJ
ap-1406	106	2	β	β	X
ap-1406	106	3	is	be	AUX
ap-1406	106	4	an	an	DET
ap-1406	106	5	algebraic	algebraic	ADJ
ap-1406	106	6	number	number	NOUN
ap-1406	106	7	of	of	ADP
ap-1406	106	8	degree	degree	NOUN
ap-1406	106	9	n	n	CCONJ
ap-1406	106	10	,	,	PUNCT
ap-1406	106	11	then	then	ADV
ap-1406	106	12	the	the	DET
ap-1406	106	13	minimal	minimal	ADJ
ap-1406	106	14	subfield	subfield	NOUN
ap-1406	106	15	of	of	ADP
ap-1406	106	16	the	the	DET
ap-1406	106	17	field	field	NOUN
ap-1406	106	18	of	of	ADP
ap-1406	106	19	complex	complex	ADJ
ap-1406	106	20	numbers	number	NOUN
ap-1406	106	21	containing	contain	VERB
ap-1406	106	22	β	β	X
ap-1406	106	23	is	be	AUX
ap-1406	106	24	denoted	denote	VERB
ap-1406	106	25	by	by	ADP
ap-1406	106	26	q(β	q(β	NOUN
ap-1406	106	27	)	)	PUNCT
ap-1406	106	28	and	and	CCONJ
ap-1406	106	29	is	be	AUX
ap-1406	106	30	of	of	ADP
ap-1406	106	31	the	the	DET
ap-1406	106	32	form	form	NOUN
ap-1406	106	33	q(β	q(β	PROPN
ap-1406	106	34	)	)	PUNCT
ap-1406	107	1	=	=	SYM
ap-1406	107	2	{	{	PUNCT
ap-1406	107	3	c0	c0	NOUN
ap-1406	107	4	+	+	CCONJ
ap-1406	107	5	c1β	c1β	PROPN
ap-1406	107	6	+	+	X
ap-1406	107	7	.	.	PUNCT
ap-1406	107	8	.	.	PUNCT
ap-1406	107	9	.	.	PUNCT
ap-1406	108	1	+	+	CCONJ
ap-1406	109	1	cn−1β	cn−1β	PROPN
ap-1406	109	2	n−1	n−1	PROPN
ap-1406	109	3	|	|	ADV
ap-1406	109	4	ci	ci	PROPN
ap-1406	109	5	∈	∈	PROPN
ap-1406	110	1	q	q	X
ap-1406	110	2	}	}	PUNCT
ap-1406	110	3	.	.	PUNCT
ap-1406	111	1	60	60	NUM
ap-1406	111	2	acta	acta	PROPN
ap-1406	111	3	polytechnica	polytechnica	PROPN
ap-1406	111	4	vol	vol	NOUN
ap-1406	111	5	.	.	PUNCT
ap-1406	112	1	51	51	NUM
ap-1406	112	2	no	no	INTJ
ap-1406	112	3	.	.	PUNCT
ap-1406	113	1	4/2011	4/2011	NUM
ap-1406	113	2	if	if	SCONJ
ap-1406	113	3	γ	γ	NOUN
ap-1406	113	4	is	be	AUX
ap-1406	113	5	a	a	DET
ap-1406	113	6	conjugate	conjugate	NOUN
ap-1406	113	7	of	of	ADP
ap-1406	113	8	an	an	DET
ap-1406	113	9	algebraic	algebraic	ADJ
ap-1406	113	10	number	number	NOUN
ap-1406	113	11	β	β	NOUN
ap-1406	113	12	,	,	PUNCT
ap-1406	113	13	then	then	ADV
ap-1406	113	14	the	the	DET
ap-1406	113	15	fields	field	NOUN
ap-1406	113	16	q(β	q(β	NOUN
ap-1406	113	17	)	)	PUNCT
ap-1406	113	18	and	and	CCONJ
ap-1406	113	19	q(γ	q(γ	PROPN
ap-1406	113	20	)	)	PUNCT
ap-1406	113	21	are	be	AUX
ap-1406	113	22	isomorphic	isomorphic	ADJ
ap-1406	113	23	.	.	PUNCT
ap-1406	114	1	the	the	DET
ap-1406	114	2	corresponding	corresponding	ADJ
ap-1406	114	3	isomorphism	isomorphism	NOUN
ap-1406	114	4	is	be	AUX
ap-1406	114	5	given	give	VERB
ap-1406	114	6	by	by	ADP
ap-1406	114	7	c0+c1β+	c0+c1β+	PUNCT
ap-1406	114	8	.	.	PUNCT
ap-1406	114	9	.	.	PUNCT
ap-1406	115	1	.+cn−1β	.+cn−1β	PUNCT
ap-1406	116	1	n−1	n−1	PROPN
ap-1406	116	2	�	�	PROPN
ap-1406	116	3	→	→	SYM
ap-1406	116	4	c0+c1γ+	c0+c1γ+	PROPN
ap-1406	116	5	.	.	PUNCT
ap-1406	116	6	.	.	PUNCT
ap-1406	117	1	.+cn−1γ	.+cn−1γ	PUNCT
ap-1406	118	1	n−1	n−1	INTJ
ap-1406	118	2	.	.	PUNCT
ap-1406	119	1	in	in	ADP
ap-1406	119	2	particular	particular	ADJ
ap-1406	119	3	,	,	PUNCT
ap-1406	119	4	this	this	PRON
ap-1406	119	5	means	mean	VERB
ap-1406	119	6	that	that	SCONJ
ap-1406	119	7	β	β	NOUN
ap-1406	119	8	is	be	AUX
ap-1406	119	9	a	a	DET
ap-1406	119	10	root	root	NOUN
ap-1406	119	11	of	of	ADP
ap-1406	119	12	some	some	DET
ap-1406	119	13	polynomial	polynomial	ADJ
ap-1406	119	14	f	f	NOUN
ap-1406	119	15	with	with	ADP
ap-1406	119	16	rational	rational	ADJ
ap-1406	119	17	coefficients	coefficient	NOUN
ap-1406	119	18	if	if	SCONJ
ap-1406	119	19	and	and	CCONJ
ap-1406	119	20	only	only	ADV
ap-1406	119	21	if	if	SCONJ
ap-1406	119	22	γ	γ	NOUN
ap-1406	119	23	is	be	AUX
ap-1406	119	24	a	a	DET
ap-1406	119	25	root	root	NOUN
ap-1406	119	26	of	of	ADP
ap-1406	119	27	the	the	DET
ap-1406	119	28	same	same	ADJ
ap-1406	119	29	polynomial	polynomial	ADJ
ap-1406	119	30	f	f	NOUN
ap-1406	119	31	.	.	PUNCT
ap-1406	120	1	3	3	NUM
ap-1406	120	2	ito	ito	PROPN
ap-1406	120	3	-	-	PUNCT
ap-1406	120	4	sadahiro	sadahiro	NOUN
ap-1406	120	5	polynomial	polynomial	NOUN
ap-1406	120	6	from	from	ADP
ap-1406	120	7	now	now	ADV
ap-1406	120	8	on	on	ADV
ap-1406	120	9	,	,	PUNCT
ap-1406	120	10	we	we	PRON
ap-1406	120	11	shall	shall	AUX
ap-1406	120	12	consider	consider	VERB
ap-1406	120	13	for	for	ADP
ap-1406	120	14	bases	basis	NOUN
ap-1406	120	15	of	of	ADP
ap-1406	120	16	the	the	DET
ap-1406	120	17	numeration	numeration	NOUN
ap-1406	120	18	system	system	NOUN
ap-1406	120	19	only	only	ADV
ap-1406	120	20	ito	ito	PROPN
ap-1406	120	21	-	-	PUNCT
ap-1406	120	22	sadahiro	sadahiro	NOUN
ap-1406	120	23	numbers	number	NOUN
ap-1406	120	24	,	,	PUNCT
ap-1406	120	25	i.e.	i.e.	X
ap-1406	120	26	numbers	number	NOUN
ap-1406	120	27	β	β	VERB
ap-1406	120	28	such	such	ADJ
ap-1406	120	29	that	that	DET
ap-1406	120	30	d−β(lβ	d−β(lβ	NOUN
ap-1406	120	31	)	)	PUNCT
ap-1406	120	32	=	=	SYM
ap-1406	120	33	d1	d1	PROPN
ap-1406	120	34	.	.	PUNCT
ap-1406	120	35	.	.	PUNCT
ap-1406	120	36	.	.	PUNCT
ap-1406	121	1	dm(dm+1	dm(dm+1	ADJ
ap-1406	121	2	.	.	PUNCT
ap-1406	121	3	.	.	PUNCT
ap-1406	121	4	.	.	PUNCT
ap-1406	122	1	dm+p)ω	dm+p)ω	PRON
ap-1406	122	2	.	.	PUNCT
ap-1406	123	1	(	(	PUNCT
ap-1406	123	2	7	7	X
ap-1406	123	3	)	)	PUNCT
ap-1406	123	4	without	without	ADP
ap-1406	123	5	loss	loss	NOUN
ap-1406	123	6	of	of	ADP
ap-1406	123	7	generality	generality	NOUN
ap-1406	123	8	we	we	PRON
ap-1406	123	9	shall	shall	AUX
ap-1406	123	10	assume	assume	VERB
ap-1406	123	11	that	that	SCONJ
ap-1406	123	12	m	m	PROPN
ap-1406	123	13	≥	≥	NOUN
ap-1406	123	14	0	0	NUM
ap-1406	123	15	,	,	PUNCT
ap-1406	123	16	p	p	PRON
ap-1406	123	17	≥	≥	NUM
ap-1406	123	18	1	1	NUM
ap-1406	123	19	are	be	AUX
ap-1406	123	20	minimal	minimal	ADJ
ap-1406	123	21	values	value	NOUN
ap-1406	123	22	so	so	SCONJ
ap-1406	123	23	that	that	PRON
ap-1406	123	24	d−β(lβ	d−β(lβ	NOUN
ap-1406	123	25	)	)	PUNCT
ap-1406	123	26	can	can	AUX
ap-1406	123	27	be	be	AUX
ap-1406	123	28	written	write	VERB
ap-1406	123	29	in	in	ADP
ap-1406	123	30	the	the	DET
ap-1406	123	31	above	above	ADJ
ap-1406	123	32	form	form	NOUN
ap-1406	123	33	.	.	PUNCT
ap-1406	124	1	recall	recall	VERB
ap-1406	124	2	that	that	PRON
ap-1406	124	3	lβ	lβ	VERB
ap-1406	124	4	=	=	PUNCT
ap-1406	124	5	−	−	PROPN
ap-1406	124	6	β	β	X
ap-1406	124	7	β	β	NOUN
ap-1406	125	1	+	+	ADP
ap-1406	125	2	1	1	NUM
ap-1406	125	3	.	.	PUNCT
ap-1406	126	1	therefore	therefore	ADV
ap-1406	126	2	(	(	PUNCT
ap-1406	126	3	7	7	X
ap-1406	126	4	)	)	PUNCT
ap-1406	126	5	can	can	AUX
ap-1406	126	6	be	be	AUX
ap-1406	126	7	rewritten	rewrite	VERB
ap-1406	126	8	as	as	ADP
ap-1406	126	9	−	−	PROPN
ap-1406	126	10	β	β	X
ap-1406	126	11	β	β	X
ap-1406	127	1	+	+	CCONJ
ap-1406	127	2	1	1	X
ap-1406	127	3	=	=	SYM
ap-1406	127	4	d1	d1	PROPN
ap-1406	127	5	−β	−β	NOUN
ap-1406	127	6	+	+	X
ap-1406	127	7	.	.	PUNCT
ap-1406	127	8	.	.	PUNCT
ap-1406	127	9	.	.	PUNCT
ap-1406	128	1	+	+	CCONJ
ap-1406	128	2	dm	dm	X
ap-1406	128	3	(	(	PUNCT
ap-1406	128	4	−β)m	−β)m	NOUN
ap-1406	128	5	+	+	PROPN
ap-1406	128	6	(	(	PUNCT
ap-1406	128	7	dm+1	dm+1	PROPN
ap-1406	128	8	(	(	PUNCT
ap-1406	128	9	−β)m+1	−β)m+1	PROPN
ap-1406	128	10	+	+	PROPN
ap-1406	128	11	.	.	PUNCT
ap-1406	128	12	.	.	PUNCT
ap-1406	128	13	.	.	PUNCT
ap-1406	129	1	+	+	PUNCT
ap-1406	129	2	dm+p	dm+p	NOUN
ap-1406	129	3	(	(	PUNCT
ap-1406	129	4	−β)m+p	−β)m+p	NOUN
ap-1406	129	5	)	)	PUNCT
ap-1406	130	1	∞∑	∞∑	PRON
ap-1406	130	2	i=0	i=0	PROPN
ap-1406	130	3	1	1	NUM
ap-1406	130	4	(	(	PUNCT
ap-1406	130	5	−β)p	−β)p	NOUN
ap-1406	130	6	i	i	PRON
ap-1406	130	7	,	,	PUNCT
ap-1406	130	8	and	and	CCONJ
ap-1406	130	9	after	after	ADP
ap-1406	130	10	arrangement	arrangement	NOUN
ap-1406	130	11	0	0	NUM
ap-1406	130	12	=	=	PUNCT
ap-1406	130	13	−β	−β	PROPN
ap-1406	130	14	−β	−β	NOUN
ap-1406	130	15	−	−	PROPN
ap-1406	130	16	1	1	NUM
ap-1406	130	17	+	+	CCONJ
ap-1406	130	18	d1	d1	PROPN
ap-1406	130	19	−β	−β	NOUN
ap-1406	130	20	+	+	X
ap-1406	130	21	.	.	PUNCT
ap-1406	130	22	.	.	PUNCT
ap-1406	130	23	.	.	PUNCT
ap-1406	131	1	+	+	CCONJ
ap-1406	131	2	dm	dm	X
ap-1406	131	3	(	(	PUNCT
ap-1406	131	4	−β)m	−β)m	NOUN
ap-1406	131	5	+	+	CCONJ
ap-1406	131	6	(	(	PUNCT
ap-1406	131	7	−β)p	−β)p	NOUN
ap-1406	131	8	(	(	PUNCT
ap-1406	131	9	−β)p	−β)p	NOUN
ap-1406	131	10	−	−	NUM
ap-1406	131	11	1	1	NUM
ap-1406	131	12	·	·	PUNCT
ap-1406	131	13	(	(	PUNCT
ap-1406	131	14	dm+1	dm+1	PROPN
ap-1406	131	15	(	(	PUNCT
ap-1406	131	16	−β)m+1	−β)m+1	PROPN
ap-1406	131	17	+	+	PROPN
ap-1406	131	18	.	.	PUNCT
ap-1406	131	19	.	.	PUNCT
ap-1406	131	20	.	.	PUNCT
ap-1406	132	1	+	+	PUNCT
ap-1406	132	2	dm+p	dm+p	NOUN
ap-1406	132	3	(	(	PUNCT
ap-1406	132	4	−β)m+p	−β)m+p	NOUN
ap-1406	132	5	)	)	PUNCT
ap-1406	132	6	.	.	PUNCT
ap-1406	133	1	multiplying	multiply	VERB
ap-1406	133	2	by	by	ADP
ap-1406	133	3	(	(	PUNCT
ap-1406	133	4	−β)m	−β)m	NOUN
ap-1406	133	5	(	(	PUNCT
ap-1406	133	6	(	(	PUNCT
ap-1406	133	7	−β)p	−β)p	PROPN
ap-1406	133	8	−1	−1	NOUN
ap-1406	133	9	)	)	PUNCT
ap-1406	133	10	,	,	PUNCT
ap-1406	133	11	we	we	PRON
ap-1406	133	12	obtain	obtain	VERB
ap-1406	133	13	the	the	DET
ap-1406	133	14	following	follow	VERB
ap-1406	133	15	lemma	lemma	PROPN
ap-1406	133	16	.	.	PUNCT
ap-1406	134	1	lemma	lemma	PROPN
ap-1406	134	2	1	1	NUM
ap-1406	134	3	let	let	VERB
ap-1406	134	4	β	β	NOUN
ap-1406	134	5	be	be	AUX
ap-1406	134	6	an	an	DET
ap-1406	134	7	ito	ito	PROPN
ap-1406	134	8	-	-	PUNCT
ap-1406	134	9	sadahiro	sadahiro	NOUN
ap-1406	134	10	number	number	NOUN
ap-1406	134	11	and	and	CCONJ
ap-1406	134	12	let	let	VERB
ap-1406	134	13	d−β(lβ	d−β(lβ	NOUN
ap-1406	134	14	)	)	PUNCT
ap-1406	134	15	be	be	AUX
ap-1406	134	16	of	of	ADP
ap-1406	134	17	the	the	DET
ap-1406	134	18	form	form	NOUN
ap-1406	134	19	(	(	PUNCT
ap-1406	134	20	7	7	NUM
ap-1406	134	21	)	)	PUNCT
ap-1406	134	22	.	.	PUNCT
ap-1406	135	1	then	then	ADV
ap-1406	135	2	β	β	PROPN
ap-1406	135	3	is	be	AUX
ap-1406	135	4	a	a	DET
ap-1406	135	5	root	root	NOUN
ap-1406	135	6	of	of	ADP
ap-1406	135	7	the	the	DET
ap-1406	135	8	polynomial	polynomial	ADJ
ap-1406	135	9	p	p	NOUN
ap-1406	135	10	(	(	PUNCT
ap-1406	135	11	x	x	NOUN
ap-1406	135	12	)	)	PUNCT
ap-1406	135	13	=	=	SYM
ap-1406	136	1	(	(	PUNCT
ap-1406	136	2	−x)m+1	−x)m+1	PROPN
ap-1406	136	3	p−1∑	p−1∑	PROPN
ap-1406	136	4	i=0	i=0	PROPN
ap-1406	136	5	(	(	PUNCT
ap-1406	136	6	−x)i	−x)i	NOUN
ap-1406	136	7	+	+	CCONJ
ap-1406	136	8	(	(	PUNCT
ap-1406	136	9	(	(	PUNCT
ap-1406	136	10	−x)p	−x)p	NOUN
ap-1406	136	11	−	−	NOUN
ap-1406	136	12	1	1	NUM
ap-1406	136	13	)	)	PUNCT
ap-1406	136	14	·	·	PUNCT
ap-1406	136	15	(	(	PUNCT
ap-1406	136	16	8)	8)	NUM
ap-1406	136	17	m∑	m∑	NOUN
ap-1406	136	18	i=1	i=1	PRON
ap-1406	136	19	di(−x)m−i	di(−x)m−i	NOUN
ap-1406	136	20	+	+	CCONJ
ap-1406	136	21	m+p∑	m+p∑	PROPN
ap-1406	136	22	i	i	NOUN
ap-1406	136	23	=	=	NOUN
ap-1406	136	24	m+1	m+1	NUM
ap-1406	136	25	di(−x)m+p−i	di(−x)m+p−i	NOUN
ap-1406	136	26	.	.	PUNCT
ap-1406	137	1	such	such	DET
ap-1406	137	2	a	a	DET
ap-1406	137	3	polynomial	polynomial	NOUN
ap-1406	137	4	is	be	AUX
ap-1406	137	5	called	call	VERB
ap-1406	137	6	the	the	DET
ap-1406	137	7	ito	ito	PROPN
ap-1406	137	8	-	-	PROPN
ap-1406	137	9	sadahiro	sadahiro	PROPN
ap-1406	137	10	polynomial	polynomial	NOUN
ap-1406	137	11	of	of	ADP
ap-1406	137	12	β	β	PROPN
ap-1406	137	13	.	.	PUNCT
ap-1406	138	1	corollary	corollary	ADJ
ap-1406	138	2	2	2	NUM
ap-1406	138	3	an	an	DET
ap-1406	138	4	ito	ito	PROPN
ap-1406	138	5	-	-	PUNCT
ap-1406	138	6	sadahiro	sadahiro	NOUN
ap-1406	138	7	number	number	NOUN
ap-1406	138	8	is	be	AUX
ap-1406	138	9	an	an	DET
ap-1406	138	10	algebraic	algebraic	ADJ
ap-1406	138	11	integer	integer	NOUN
ap-1406	138	12	of	of	ADP
ap-1406	138	13	degree	degree	NOUN
ap-1406	138	14	smaller	small	ADJ
ap-1406	138	15	than	than	ADP
ap-1406	138	16	or	or	CCONJ
ap-1406	138	17	equal	equal	ADJ
ap-1406	138	18	to	to	ADP
ap-1406	138	19	m+p	m+p	NOUN
ap-1406	138	20	,	,	PUNCT
ap-1406	138	21	where	where	SCONJ
ap-1406	138	22	m	m	X
ap-1406	138	23	,	,	PUNCT
ap-1406	138	24	p	p	NOUN
ap-1406	138	25	are	be	AUX
ap-1406	138	26	given	give	VERB
ap-1406	138	27	by	by	ADP
ap-1406	138	28	(	(	PUNCT
ap-1406	138	29	7	7	NUM
ap-1406	138	30	)	)	PUNCT
ap-1406	138	31	.	.	PUNCT
ap-1406	139	1	it	it	PRON
ap-1406	139	2	is	be	AUX
ap-1406	139	3	useful	useful	ADJ
ap-1406	139	4	to	to	PART
ap-1406	139	5	mention	mention	VERB
ap-1406	139	6	that	that	SCONJ
ap-1406	139	7	the	the	DET
ap-1406	139	8	ito	ito	PROPN
ap-1406	139	9	-	-	PUNCT
ap-1406	139	10	sadahiro	sadahiro	PROPN
ap-1406	139	11	polynomial	polynomial	NOUN
ap-1406	139	12	is	be	AUX
ap-1406	139	13	not	not	PART
ap-1406	139	14	necessarily	necessarily	ADV
ap-1406	139	15	irreducible	irreducible	ADJ
ap-1406	139	16	over	over	ADP
ap-1406	139	17	q.	q.	PROPN
ap-1406	139	18	as	as	ADP
ap-1406	139	19	an	an	DET
ap-1406	139	20	example	example	NOUN
ap-1406	139	21	one	one	PRON
ap-1406	139	22	can	can	AUX
ap-1406	139	23	take	take	VERB
ap-1406	139	24	the	the	DET
ap-1406	139	25	minimal	minimal	ADJ
ap-1406	139	26	pisot	pisot	ADJ
ap-1406	139	27	number	number	NOUN
ap-1406	139	28	.	.	PUNCT
ap-1406	140	1	for	for	ADP
ap-1406	140	2	such	such	ADJ
ap-1406	140	3	β	β	NOUN
ap-1406	140	4	,	,	PUNCT
ap-1406	140	5	we	we	PRON
ap-1406	140	6	have	have	VERB
ap-1406	140	7	d−β(lβ	d−β(lβ	NOUN
ap-1406	140	8	)	)	PUNCT
ap-1406	140	9	=	=	SYM
ap-1406	140	10	1	1	NUM
ap-1406	140	11	001ω	001ω	NUM
ap-1406	140	12	,	,	PUNCT
ap-1406	140	13	and	and	CCONJ
ap-1406	140	14	thus	thus	ADV
ap-1406	140	15	the	the	DET
ap-1406	140	16	ito	ito	PROPN
ap-1406	140	17	-	-	PROPN
ap-1406	140	18	sadahiro	sadahiro	PROPN
ap-1406	140	19	polynomial	polynomial	NOUN
ap-1406	140	20	is	be	AUX
ap-1406	140	21	equal	equal	ADJ
ap-1406	140	22	to	to	ADP
ap-1406	140	23	p	p	PROPN
ap-1406	140	24	(	(	PUNCT
ap-1406	140	25	x	x	NOUN
ap-1406	140	26	)	)	PUNCT
ap-1406	140	27	=	=	SYM
ap-1406	141	1	x4−x3−x2	x4−x3−x2	X
ap-1406	141	2	+	+	PROPN
ap-1406	141	3	1	1	NUM
ap-1406	141	4	=	=	SYM
ap-1406	141	5	(	(	PUNCT
ap-1406	141	6	x−1)(x3−x−1	x−1)(x3−x−1	PROPN
ap-1406	141	7	)	)	PUNCT
ap-1406	141	8	,	,	PUNCT
ap-1406	141	9	where	where	SCONJ
ap-1406	141	10	x3−x−1	x3−x−1	PROPN
ap-1406	141	11	is	be	AUX
ap-1406	141	12	the	the	DET
ap-1406	141	13	minimal	minimal	ADJ
ap-1406	141	14	polynomial	polynomial	NOUN
ap-1406	141	15	of	of	ADP
ap-1406	141	16	β	β	PROPN
ap-1406	141	17	.	.	PUNCT
ap-1406	141	18	remark	remark	PROPN
ap-1406	141	19	3	3	NUM
ap-1406	141	20	note	note	VERB
ap-1406	141	21	that	that	SCONJ
ap-1406	141	22	for	for	ADP
ap-1406	141	23	p	p	NOUN
ap-1406	141	24	=	=	SYM
ap-1406	141	25	1	1	NUM
ap-1406	141	26	and	and	CCONJ
ap-1406	141	27	dm+1	dm+1	PROPN
ap-1406	141	28	=	=	SYM
ap-1406	141	29	0	0	NUM
ap-1406	141	30	,	,	PUNCT
ap-1406	141	31	we	we	PRON
ap-1406	141	32	have	have	VERB
ap-1406	141	33	d−β(lβ	d−β(lβ	NOUN
ap-1406	141	34	)	)	PUNCT
ap-1406	142	1	=	=	SYM
ap-1406	142	2	d1	d1	PROPN
ap-1406	142	3	.	.	PUNCT
ap-1406	142	4	.	.	PUNCT
ap-1406	142	5	.	.	PUNCT
ap-1406	143	1	dm0ω	dm0ω	ADV
ap-1406	143	2	,	,	PUNCT
ap-1406	143	3	and	and	CCONJ
ap-1406	143	4	the	the	DET
ap-1406	143	5	ito	ito	PROPN
ap-1406	143	6	-	-	PROPN
ap-1406	143	7	sadahiro	sadahiro	PROPN
ap-1406	143	8	polynomial	polynomial	NOUN
ap-1406	143	9	of	of	ADP
ap-1406	143	10	β	β	PROPN
ap-1406	143	11	is	be	AUX
ap-1406	143	12	of	of	ADP
ap-1406	143	13	the	the	DET
ap-1406	143	14	form	form	NOUN
ap-1406	143	15	p	p	X
ap-1406	143	16	(	(	PUNCT
ap-1406	143	17	x	x	NOUN
ap-1406	143	18	)	)	PUNCT
ap-1406	143	19	=	=	SYM
ap-1406	143	20	(	(	PUNCT
ap-1406	143	21	−x)m+1	−x)m+1	PROPN
ap-1406	143	22	+	+	CCONJ
ap-1406	143	23	d1(−x)m	d1(−x)m	PROPN
ap-1406	143	24	+	+	CCONJ
ap-1406	143	25	(	(	PUNCT
ap-1406	143	26	d2	d2	PROPN
ap-1406	143	27	−	−	PROPN
ap-1406	143	28	d1)(−x)m−1	d1)(−x)m−1	PROPN
ap-1406	143	29	+	+	X
ap-1406	143	30	.	.	PUNCT
ap-1406	143	31	.	.	PUNCT
ap-1406	143	32	.	.	PUNCT
ap-1406	144	1	+	+	CCONJ
ap-1406	144	2	(	(	PUNCT
ap-1406	144	3	dm	dm	X
ap-1406	144	4	−	−	PROPN
ap-1406	144	5	dm−1)(−x	dm−1)(−x	PROPN
ap-1406	144	6	)	)	PUNCT
ap-1406	145	1	−	−	PROPN
ap-1406	146	1	dm	dm	INTJ
ap-1406	146	2	,	,	PUNCT
ap-1406	146	3	(	(	PUNCT
ap-1406	146	4	9	9	NUM
ap-1406	146	5	)	)	PUNCT
ap-1406	146	6	and	and	CCONJ
ap-1406	146	7	thus	thus	ADV
ap-1406	146	8	β	β	X
ap-1406	146	9	is	be	AUX
ap-1406	146	10	an	an	DET
ap-1406	146	11	algebraic	algebraic	ADJ
ap-1406	146	12	integer	integer	NOUN
ap-1406	146	13	of	of	ADP
ap-1406	146	14	degree	degree	NOUN
ap-1406	146	15	at	at	ADP
ap-1406	146	16	most	most	ADV
ap-1406	146	17	m	m	VERB
ap-1406	146	18	+	+	ADJ
ap-1406	146	19	1	1	NUM
ap-1406	146	20	.	.	X
ap-1406	146	21	theorem	theorem	NOUN
ap-1406	146	22	4	4	NUM
ap-1406	146	23	let	let	VERB
ap-1406	146	24	β	β	NOUN
ap-1406	146	25	be	be	AUX
ap-1406	146	26	an	an	DET
ap-1406	146	27	ito	ito	PROPN
ap-1406	146	28	-	-	PUNCT
ap-1406	146	29	sadahiro	sadahiro	NOUN
ap-1406	146	30	number	number	NOUN
ap-1406	146	31	.	.	PUNCT
ap-1406	147	1	all	all	DET
ap-1406	147	2	roots	root	NOUN
ap-1406	147	3	γ	γ	PROPN
ap-1406	147	4	,	,	PUNCT
ap-1406	147	5	γ	γ	PROPN
ap-1406	147	6	�	�	PROPN
ap-1406	147	7	=	=	SYM
ap-1406	147	8	β	β	NOUN
ap-1406	147	9	,	,	PUNCT
ap-1406	147	10	of	of	ADP
ap-1406	147	11	the	the	DET
ap-1406	147	12	ito	ito	PROPN
ap-1406	147	13	-	-	PROPN
ap-1406	147	14	sadahiro	sadahiro	PROPN
ap-1406	147	15	polynomial	polynomial	NOUN
ap-1406	147	16	(	(	PUNCT
ap-1406	147	17	in	in	ADP
ap-1406	147	18	particular	particular	ADJ
ap-1406	147	19	all	all	DET
ap-1406	147	20	conjugates	conjugate	NOUN
ap-1406	147	21	of	of	ADP
ap-1406	147	22	β	β	NOUN
ap-1406	147	23	)	)	PUNCT
ap-1406	147	24	satisfy	satisfy	VERB
ap-1406	147	25	|γ|	|γ|	ADV
ap-1406	147	26	<	<	X
ap-1406	147	27	2	2	NUM
ap-1406	147	28	.	.	PUNCT
ap-1406	148	1	proof	proof	NOUN
ap-1406	148	2	.	.	PUNCT
ap-1406	149	1	since	since	SCONJ
ap-1406	149	2	β	β	X
ap-1406	149	3	is	be	AUX
ap-1406	149	4	a	a	DET
ap-1406	149	5	root	root	NOUN
ap-1406	149	6	of	of	ADP
ap-1406	149	7	its	its	PRON
ap-1406	149	8	ito	ito	PROPN
ap-1406	149	9	-	-	PROPN
ap-1406	149	10	sadahiro	sadahiro	PROPN
ap-1406	149	11	polynomial	polynomial	PROPN
ap-1406	149	12	p	p	PROPN
ap-1406	149	13	,	,	PUNCT
ap-1406	149	14	there	there	PRON
ap-1406	149	15	must	must	AUX
ap-1406	149	16	exist	exist	VERB
ap-1406	149	17	a	a	DET
ap-1406	149	18	polynomial	polynomial	ADJ
ap-1406	149	19	q	q	NOUN
ap-1406	149	20	such	such	ADJ
ap-1406	149	21	that	that	SCONJ
ap-1406	149	22	p	p	X
ap-1406	149	23	(	(	PUNCT
ap-1406	149	24	x	x	NOUN
ap-1406	149	25	)	)	PUNCT
ap-1406	149	26	=	=	SYM
ap-1406	149	27	(	(	PUNCT
ap-1406	149	28	x	x	X
ap-1406	149	29	−	−	PROPN
ap-1406	149	30	β)q(x	β)q(x	NUM
ap-1406	149	31	)	)	PUNCT
ap-1406	149	32	.	.	PUNCT
ap-1406	150	1	let	let	VERB
ap-1406	150	2	us	we	PRON
ap-1406	150	3	first	first	ADV
ap-1406	150	4	determine	determine	VERB
ap-1406	150	5	q	q	PUNCT
ap-1406	150	6	and	and	CCONJ
ap-1406	150	7	show	show	VERB
ap-1406	150	8	that	that	SCONJ
ap-1406	150	9	it	it	PRON
ap-1406	150	10	is	be	AUX
ap-1406	150	11	a	a	DET
ap-1406	150	12	monic	monic	ADJ
ap-1406	150	13	polynomial	polynomial	NOUN
ap-1406	150	14	with	with	ADP
ap-1406	150	15	coefficients	coefficient	NOUN
ap-1406	150	16	in	in	ADP
ap-1406	150	17	modulus	modulus	NOUN
ap-1406	150	18	not	not	PART
ap-1406	150	19	exceeding	exceed	VERB
ap-1406	150	20	1	1	NUM
ap-1406	150	21	.	.	PUNCT
ap-1406	151	1	the	the	DET
ap-1406	151	2	coefficients	coefficient	NOUN
ap-1406	151	3	di	di	VERB
ap-1406	151	4	in	in	ADP
ap-1406	151	5	the	the	DET
ap-1406	151	6	polynomial	polynomial	ADJ
ap-1406	151	7	p	p	NOUN
ap-1406	151	8	in	in	ADP
ap-1406	151	9	the	the	DET
ap-1406	151	10	form	form	NOUN
ap-1406	151	11	(	(	PUNCT
ap-1406	151	12	8)	8)	NUM
ap-1406	151	13	are	be	AUX
ap-1406	151	14	the	the	DET
ap-1406	151	15	digits	digit	NOUN
ap-1406	151	16	of	of	ADP
ap-1406	151	17	the	the	DET
ap-1406	151	18	(	(	PUNCT
ap-1406	151	19	−β)-expansion	−β)-expansion	NOUN
ap-1406	151	20	of	of	ADP
ap-1406	151	21	lβ	lβ	NOUN
ap-1406	151	22	,	,	PUNCT
ap-1406	151	23	and	and	CCONJ
ap-1406	151	24	thus	thus	ADV
ap-1406	151	25	,	,	PUNCT
ap-1406	151	26	by	by	ADP
ap-1406	151	27	(	(	PUNCT
ap-1406	151	28	5	5	NUM
ap-1406	151	29	)	)	PUNCT
ap-1406	151	30	,	,	PUNCT
ap-1406	151	31	they	they	PRON
ap-1406	151	32	satisfy	satisfy	VERB
ap-1406	151	33	di	di	NOUN
ap-1406	151	34	=	=	NOUN
ap-1406	151	35	#	#	SYM
ap-1406	151	36	−βt	−βt	NOUN
ap-1406	151	37	i−1(lβ	i−1(lβ	NOUN
ap-1406	151	38	)	)	PUNCT
ap-1406	152	1	−	−	ADP
ap-1406	152	2	lβ$.	lβ$.	NOUN
ap-1406	152	3	relation	relation	NOUN
ap-1406	152	4	(	(	PUNCT
ap-1406	152	5	4	4	NUM
ap-1406	152	6	)	)	PUNCT
ap-1406	152	7	then	then	ADV
ap-1406	152	8	implies	imply	VERB
ap-1406	152	9	t	t	NOUN
ap-1406	152	10	i(lβ	i(lβ	NOUN
ap-1406	152	11	)	)	PUNCT
ap-1406	152	12	=	=	SYM
ap-1406	152	13	−βt	−βt	X
ap-1406	152	14	i−1(lβ)−#−βt	i−1(lβ)−#−βt	PROPN
ap-1406	152	15	i−1(lβ)−lβ$	i−1(lβ)−lβ$	PROPN
ap-1406	152	16	,	,	PUNCT
ap-1406	152	17	wherefrom	wherefrom	ADP
ap-1406	152	18	we	we	PRON
ap-1406	152	19	have	have	VERB
ap-1406	152	20	di	di	NOUN
ap-1406	152	21	=	=	NOUN
ap-1406	152	22	−t	−t	PROPN
ap-1406	152	23	i(lβ	i(lβ	NOUN
ap-1406	152	24	)	)	PUNCT
ap-1406	152	25	−	−	NOUN
ap-1406	152	26	βt	βt	NUM
ap-1406	152	27	i−1(lβ	i−1(lβ	NUM
ap-1406	152	28	)	)	PUNCT
ap-1406	152	29	.	.	PUNCT
ap-1406	153	1	for	for	ADP
ap-1406	153	2	simplicity	simplicity	NOUN
ap-1406	153	3	of	of	ADP
ap-1406	153	4	notation	notation	NOUN
ap-1406	153	5	in	in	ADP
ap-1406	153	6	this	this	DET
ap-1406	153	7	proof	proof	NOUN
ap-1406	153	8	,	,	PUNCT
ap-1406	153	9	denote	denote	VERB
ap-1406	153	10	ti	ti	PROPN
ap-1406	153	11	=	=	SYM
ap-1406	153	12	t	t	PROPN
ap-1406	153	13	i(lβ	i(lβ	NOUN
ap-1406	153	14	)	)	PUNCT
ap-1406	153	15	,	,	PUNCT
ap-1406	153	16	for	for	ADP
ap-1406	153	17	i	i	PROPN
ap-1406	153	18	=	=	SYM
ap-1406	153	19	0	0	NUM
ap-1406	153	20	,	,	PUNCT
ap-1406	153	21	1	1	NUM
ap-1406	153	22	,	,	PUNCT
ap-1406	153	23	.	.	PUNCT
ap-1406	153	24	.	.	PUNCT
ap-1406	154	1	.	.	PUNCT
ap-1406	155	1	,	,	PUNCT
ap-1406	155	2	m	m	VERB
ap-1406	155	3	+	+	X
ap-1406	155	4	p.	p.	NOUN
ap-1406	155	5	substituting	substitute	VERB
ap-1406	155	6	di	di	NOUN
ap-1406	155	7	=	=	PUNCT
ap-1406	155	8	−ti	−ti	NOUN
ap-1406	156	1	−	−	PROPN
ap-1406	156	2	βti−1	βti−1	PROPN
ap-1406	156	3	into	into	ADP
ap-1406	156	4	(	(	PUNCT
ap-1406	156	5	8)	8)	NUM
ap-1406	156	6	,	,	PUNCT
ap-1406	156	7	we	we	PRON
ap-1406	156	8	obtain	obtain	VERB
ap-1406	156	9	p	p	NOUN
ap-1406	156	10	(	(	PUNCT
ap-1406	156	11	x	x	NOUN
ap-1406	156	12	)	)	PUNCT
ap-1406	156	13	=	=	SYM
ap-1406	157	1	(	(	PUNCT
ap-1406	157	2	−x)m+1	−x)m+1	PROPN
ap-1406	157	3	p−1∑	p−1∑	PROPN
ap-1406	157	4	i=0	i=0	PROPN
ap-1406	157	5	(	(	PUNCT
ap-1406	157	6	−x)i	−x)i	NOUN
ap-1406	157	7	+	+	CCONJ
ap-1406	157	8	(	(	PUNCT
ap-1406	157	9	(	(	PUNCT
ap-1406	157	10	−x)p	−x)p	NOUN
ap-1406	157	11	−	−	NOUN
ap-1406	157	12	1	1	NUM
ap-1406	157	13	)	)	PUNCT
ap-1406	157	14	·	·	PUNCT
ap-1406	157	15	m∑	m∑	CCONJ
ap-1406	157	16	i=1	i=1	PROPN
ap-1406	157	17	(	(	PUNCT
ap-1406	157	18	−ti	−ti	NOUN
ap-1406	157	19	−	−	NOUN
ap-1406	157	20	βti−1)(−x)m−i	βti−1)(−x)m−i	NOUN
ap-1406	157	21	+	+	CCONJ
ap-1406	157	22	m+p∑	m+p∑	PROPN
ap-1406	157	23	i	i	NOUN
ap-1406	157	24	=	=	NOUN
ap-1406	157	25	m+1	m+1	X
ap-1406	157	26	(	(	PUNCT
ap-1406	157	27	−ti	−ti	NOUN
ap-1406	157	28	−	−	PROPN
ap-1406	157	29	βti−1)(−x)m+p−i	βti−1)(−x)m+p−i	PUNCT
ap-1406	157	30	=	=	PUNCT
ap-1406	157	31	(	(	PUNCT
ap-1406	157	32	−x)m+1	−x)m+1	PROPN
ap-1406	157	33	p−1∑	p−1∑	PROPN
ap-1406	157	34	i=0	i=0	PROPN
ap-1406	157	35	(	(	PUNCT
ap-1406	157	36	−x)i	−x)i	NOUN
ap-1406	157	37	+	+	CCONJ
ap-1406	157	38	(	(	PUNCT
ap-1406	157	39	(	(	PUNCT
ap-1406	157	40	−x)p	−x)p	NOUN
ap-1406	157	41	−	−	NOUN
ap-1406	157	42	1	1	NUM
ap-1406	157	43	)	)	PUNCT
ap-1406	157	44	(	(	PUNCT
ap-1406	157	45	x	x	X
ap-1406	157	46	−	−	NOUN
ap-1406	157	47	β	β	X
ap-1406	157	48	)	)	PUNCT
ap-1406	157	49	·	·	PUNCT
ap-1406	157	50	m∑	m∑	CCONJ
ap-1406	157	51	i=2	i=2	PROPN
ap-1406	157	52	ti−1(−x)m−i	ti−1(−x)m−i	X
ap-1406	157	53	+	+	X
ap-1406	157	54	(	(	PUNCT
ap-1406	157	55	10	10	NUM
ap-1406	157	56	)	)	PUNCT
ap-1406	157	57	(	(	PUNCT
ap-1406	157	58	x	x	X
ap-1406	157	59	−	−	NOUN
ap-1406	157	60	β	β	X
ap-1406	157	61	)	)	PUNCT
ap-1406	157	62	p∑	p∑	X
ap-1406	157	63	i=1	i=1	X
ap-1406	157	64	tm+i−1(−x)p−i	tm+i−1(−x)p−i	PUNCT
ap-1406	157	65	−	−	PROPN
ap-1406	157	66	(	(	PUNCT
ap-1406	157	67	(	(	PUNCT
ap-1406	157	68	−x)p	−x)p	NOUN
ap-1406	157	69	−	−	NOUN
ap-1406	157	70	1	1	NUM
ap-1406	157	71	)	)	PUNCT
ap-1406	157	72	βt0(−x)m−1	βt0(−x)m−1	PUNCT
ap-1406	158	1	+	+	CCONJ
ap-1406	158	2	tm	tm	NOUN
ap-1406	158	3	−	−	PROPN
ap-1406	158	4	tm+p	tm+p	PROPN
ap-1406	158	5	.	.	PUNCT
ap-1406	159	1	first	first	ADV
ap-1406	159	2	realize	realize	VERB
ap-1406	159	3	that	that	SCONJ
ap-1406	159	4	tm	tm	NOUN
ap-1406	159	5	−	−	PROPN
ap-1406	159	6	tm+p	tm+p	X
ap-1406	159	7	=	=	SYM
ap-1406	159	8	0	0	NUM
ap-1406	159	9	,	,	PUNCT
ap-1406	159	10	since	since	SCONJ
ap-1406	159	11	d−β(lβ	d−β(lβ	NOUN
ap-1406	159	12	)	)	PUNCT
ap-1406	159	13	is	be	AUX
ap-1406	159	14	eventually	eventually	ADV
ap-1406	159	15	periodic	periodic	ADJ
ap-1406	159	16	with	with	ADP
ap-1406	159	17	a	a	DET
ap-1406	159	18	preperiod	preperiod	NOUN
ap-1406	159	19	of	of	ADP
ap-1406	159	20	length	length	NOUN
ap-1406	159	21	m	m	PROPN
ap-1406	159	22	and	and	CCONJ
ap-1406	159	23	a	a	DET
ap-1406	159	24	period	period	NOUN
ap-1406	159	25	of	of	ADP
ap-1406	159	26	length	length	NOUN
ap-1406	159	27	p.	p.	NOUN
ap-1406	159	28	as	as	ADP
ap-1406	159	29	t0	t0	PROPN
ap-1406	159	30	=	=	SYM
ap-1406	159	31	t	t	PROPN
ap-1406	159	32	0(lβ	0(lβ	NOUN
ap-1406	159	33	)	)	PUNCT
ap-1406	160	1	=	=	SYM
ap-1406	161	1	−	−	NOUN
ap-1406	161	2	β	β	X
ap-1406	161	3	β	β	NOUN
ap-1406	162	1	+	+	ADP
ap-1406	162	2	1	1	NUM
ap-1406	162	3	,	,	PUNCT
ap-1406	162	4	we	we	PRON
ap-1406	162	5	can	can	AUX
ap-1406	162	6	derive	derive	VERB
ap-1406	162	7	that	that	SCONJ
ap-1406	162	8	(	(	PUNCT
ap-1406	162	9	−x)m+1	−x)m+1	PROPN
ap-1406	162	10	p−1∑	p−1∑	PROPN
ap-1406	162	11	i=0	i=0	PROPN
ap-1406	162	12	(	(	PUNCT
ap-1406	162	13	−x)i	−x)i	NOUN
ap-1406	162	14	−	−	PROPN
ap-1406	162	15	(	(	PUNCT
ap-1406	162	16	(	(	PUNCT
ap-1406	162	17	−x)p	−x)p	NOUN
ap-1406	162	18	−	−	NOUN
ap-1406	162	19	1	1	NUM
ap-1406	162	20	)	)	PUNCT
ap-1406	162	21	βt0(−x)m−1	βt0(−x)m−1	PUNCT
ap-1406	163	1	=	=	PUNCT
ap-1406	163	2	(	(	PUNCT
ap-1406	163	3	−x)m−1(x	−x)m−1(x	PROPN
ap-1406	163	4	−	−	PROPN
ap-1406	163	5	β)(x	β)(x	PUNCT
ap-1406	163	6	−	−	PROPN
ap-1406	163	7	t0	t0	NOUN
ap-1406	163	8	)	)	PUNCT
ap-1406	163	9	p−1∑	p−1∑	PROPN
ap-1406	163	10	i=0	i=0	PROPN
ap-1406	163	11	(	(	PUNCT
ap-1406	163	12	−x)i	−x)i	NOUN
ap-1406	163	13	.	.	PUNCT
ap-1406	164	1	61	61	NUM
ap-1406	164	2	acta	acta	PROPN
ap-1406	164	3	polytechnica	polytechnica	PROPN
ap-1406	164	4	vol	vol	NOUN
ap-1406	164	5	.	.	PUNCT
ap-1406	165	1	51	51	NUM
ap-1406	165	2	no	no	INTJ
ap-1406	165	3	.	.	PUNCT
ap-1406	166	1	4/2011	4/2011	NUM
ap-1406	166	2	putting	put	VERB
ap-1406	166	3	back	back	ADV
ap-1406	166	4	to	to	ADP
ap-1406	166	5	(	(	PUNCT
ap-1406	166	6	10	10	NUM
ap-1406	166	7	)	)	PUNCT
ap-1406	166	8	,	,	PUNCT
ap-1406	166	9	we	we	PRON
ap-1406	166	10	obtain	obtain	VERB
ap-1406	166	11	that	that	SCONJ
ap-1406	166	12	the	the	DET
ap-1406	166	13	desired	desire	VERB
ap-1406	166	14	polynomial	polynomial	NOUN
ap-1406	166	15	q	q	NOUN
ap-1406	166	16	defined	define	VERB
ap-1406	166	17	by	by	ADP
ap-1406	166	18	p	p	PROPN
ap-1406	166	19	(	(	PUNCT
ap-1406	166	20	x	x	NOUN
ap-1406	166	21	)	)	PUNCT
ap-1406	166	22	=	=	SYM
ap-1406	166	23	(	(	PUNCT
ap-1406	166	24	x	x	X
ap-1406	166	25	−	−	NOUN
ap-1406	166	26	β)q(x	β)q(x	NOUN
ap-1406	166	27	)	)	PUNCT
ap-1406	166	28	is	be	AUX
ap-1406	166	29	of	of	ADP
ap-1406	166	30	the	the	DET
ap-1406	166	31	form	form	NOUN
ap-1406	166	32	q(x	q(x	PROPN
ap-1406	166	33	)	)	PUNCT
ap-1406	166	34	=	=	PUNCT
ap-1406	167	1	(	(	PUNCT
ap-1406	167	2	−x)m−1(x	−x)m−1(x	PROPN
ap-1406	167	3	−	−	PROPN
ap-1406	167	4	t0	t0	PROPN
ap-1406	167	5	)	)	PUNCT
ap-1406	167	6	p−1∑	p−1∑	PROPN
ap-1406	167	7	i=0	i=0	PROPN
ap-1406	167	8	(	(	PUNCT
ap-1406	167	9	−x)i	−x)i	NOUN
ap-1406	167	10	+	+	CCONJ
ap-1406	167	11	(	(	PUNCT
ap-1406	167	12	(	(	PUNCT
ap-1406	167	13	−x)p	−x)p	NOUN
ap-1406	167	14	−	−	NOUN
ap-1406	167	15	1	1	NUM
ap-1406	167	16	)	)	PUNCT
ap-1406	167	17	m∑	m∑	CCONJ
ap-1406	167	18	i=2	i=2	PROPN
ap-1406	167	19	ti−1(−x)m−i	ti−1(−x)m−i	PROPN
ap-1406	167	20	+	+	NUM
ap-1406	167	21	p∑	p∑	NOUN
ap-1406	167	22	i=1	i=1	X
ap-1406	167	23	tm+i−1(−x)p−i	tm+i−1(−x)p−i	PUNCT
ap-1406	167	24	,	,	PUNCT
ap-1406	167	25	which	which	PRON
ap-1406	167	26	can	can	AUX
ap-1406	167	27	be	be	AUX
ap-1406	167	28	rewritten	rewrite	VERB
ap-1406	167	29	in	in	ADP
ap-1406	167	30	another	another	DET
ap-1406	167	31	form	form	NOUN
ap-1406	167	32	,	,	PUNCT
ap-1406	167	33	namely	namely	ADV
ap-1406	167	34	,	,	PUNCT
ap-1406	167	35	q(x	q(x	PROPN
ap-1406	167	36	)	)	PUNCT
ap-1406	168	1	=	=	SYM
ap-1406	168	2	−(−x)m+p−1	−(−x)m+p−1	PROPN
ap-1406	169	1	+	+	CCONJ
ap-1406	169	2	m+p−2∑	m+p−2∑	ADV
ap-1406	170	1	i	i	PRON
ap-1406	170	2	=	=	VERB
ap-1406	170	3	m	m	PROPN
ap-1406	170	4	(	(	PUNCT
ap-1406	170	5	tm+p−1−i	tm+p−1−i	PROPN
ap-1406	170	6	−	−	PROPN
ap-1406	170	7	t0	t0	PROPN
ap-1406	170	8	−	−	PROPN
ap-1406	170	9	1)(−x)i	1)(−x)i	NUM
ap-1406	171	1	+	+	CCONJ
ap-1406	171	2	(	(	PUNCT
ap-1406	171	3	11	11	NUM
ap-1406	171	4	)	)	PUNCT
ap-1406	171	5	m−1∑	m−1∑	PROPN
ap-1406	171	6	i=0	i=0	PROPN
ap-1406	171	7	(	(	PUNCT
ap-1406	171	8	tm+p−1−i	tm+p−1−i	NOUN
ap-1406	171	9	−	−	PROPN
ap-1406	171	10	tm−1−i)(−x)i	tm−1−i)(−x)i	PROPN
ap-1406	171	11	.	.	PUNCT
ap-1406	171	12	note	note	VERB
ap-1406	171	13	that	that	SCONJ
ap-1406	171	14	the	the	DET
ap-1406	171	15	coefficients	coefficient	NOUN
ap-1406	171	16	at	at	ADP
ap-1406	171	17	individual	individual	ADJ
ap-1406	171	18	powers	power	NOUN
ap-1406	171	19	of	of	ADP
ap-1406	171	20	−x	−x	NOUN
ap-1406	171	21	are	be	AUX
ap-1406	171	22	of	of	ADP
ap-1406	171	23	two	two	NUM
ap-1406	171	24	types	type	NOUN
ap-1406	171	25	,	,	PUNCT
ap-1406	171	26	namely	namely	ADV
ap-1406	171	27	tm+p−1−i	tm+p−1−i	VERB
ap-1406	171	28	−	−	PROPN
ap-1406	171	29	t0	t0	NOUN
ap-1406	171	30	−	−	NOUN
ap-1406	171	31	1	1	NUM
ap-1406	171	32	∈	∈	PROPN
ap-1406	172	1	[	[	X
ap-1406	172	2	−1	−1	NOUN
ap-1406	172	3	,	,	PUNCT
ap-1406	172	4	0	0	NUM
ap-1406	172	5	)	)	PUNCT
ap-1406	172	6	,	,	PUNCT
ap-1406	172	7	and	and	CCONJ
ap-1406	172	8	tm+p−1−i	tm+p−1−i	VERB
ap-1406	172	9	−	−	PROPN
ap-1406	172	10	tm−1−i	tm−1−i	NOUN
ap-1406	172	11	∈	∈	NOUN
ap-1406	172	12	(	(	PUNCT
ap-1406	172	13	−1	−1	NOUN
ap-1406	172	14	,	,	PUNCT
ap-1406	172	15	1	1	NUM
ap-1406	172	16	)	)	PUNCT
ap-1406	172	17	.	.	PUNCT
ap-1406	173	1	in	in	ADP
ap-1406	173	2	order	order	NOUN
ap-1406	173	3	to	to	PART
ap-1406	173	4	complete	complete	VERB
ap-1406	173	5	the	the	DET
ap-1406	173	6	proof	proof	NOUN
ap-1406	173	7	,	,	PUNCT
ap-1406	173	8	realize	realize	VERB
ap-1406	173	9	that	that	SCONJ
ap-1406	173	10	every	every	DET
ap-1406	173	11	root	root	NOUN
ap-1406	173	12	γ	γ	PROPN
ap-1406	173	13	,	,	PUNCT
ap-1406	173	14	γ	γ	PROPN
ap-1406	173	15	�	�	PROPN
ap-1406	173	16	=	=	SYM
ap-1406	173	17	β	β	NOUN
ap-1406	173	18	,	,	PUNCT
ap-1406	173	19	of	of	ADP
ap-1406	173	20	the	the	DET
ap-1406	173	21	polynomial	polynomial	ADJ
ap-1406	173	22	p	p	NOUN
ap-1406	173	23	satisfies	satisfie	NOUN
ap-1406	173	24	q(γ	q(γ	PROPN
ap-1406	173	25	)	)	PUNCT
ap-1406	173	26	=	=	PUNCT
ap-1406	174	1	0	0	X
ap-1406	174	2	.	.	PUNCT
ap-1406	175	1	we	we	PRON
ap-1406	175	2	thus	thus	ADV
ap-1406	175	3	have	have	VERB
ap-1406	175	4	(	(	PUNCT
ap-1406	175	5	−γ)m+p−1	−γ)m+p−1	NOUN
ap-1406	175	6	=	=	SYM
ap-1406	175	7	m+p−2∑	m+p−2∑	PROPN
ap-1406	176	1	i	i	PRON
ap-1406	176	2	=	=	VERB
ap-1406	176	3	m	m	PROPN
ap-1406	176	4	(	(	PUNCT
ap-1406	176	5	tm+p−1−i	tm+p−1−i	PROPN
ap-1406	176	6	−	−	PROPN
ap-1406	176	7	t0	t0	PROPN
ap-1406	176	8	−	−	PROPN
ap-1406	176	9	1)(−γ)i	1)(−γ)i	NUM
ap-1406	177	1	+	+	CCONJ
ap-1406	177	2	m−1∑	m−1∑	PROPN
ap-1406	177	3	i=0	i=0	PROPN
ap-1406	177	4	(	(	PUNCT
ap-1406	177	5	tm+p−1−i	tm+p−1−i	NOUN
ap-1406	177	6	−	−	PROPN
ap-1406	177	7	tm−1−i)(−γ)i	tm−1−i)(−γ)i	NOUN
ap-1406	177	8	,	,	PUNCT
ap-1406	177	9	and	and	CCONJ
ap-1406	177	10	hence	hence	ADV
ap-1406	177	11	|γ|m+p−1	|γ|m+p−1	VERB
ap-1406	177	12	≤	≤	NOUN
ap-1406	177	13	m+p−2∑	m+p−2∑	ADV
ap-1406	177	14	i=0	i=0	PROPN
ap-1406	177	15	|γ|i	|γ|i	PROPN
ap-1406	177	16	=	=	PUNCT
ap-1406	177	17	|γ|m+p−1	|γ|m+p−1	PROPN
ap-1406	177	18	−	−	NOUN
ap-1406	177	19	1	1	NUM
ap-1406	178	1	|γ|	|γ|	NOUN
ap-1406	178	2	−	−	ADP
ap-1406	178	3	1	1	NUM
ap-1406	178	4	<	<	X
ap-1406	178	5	|γ|m+p−1	|γ|m+p−1	VERB
ap-1406	178	6	|γ|	|γ|	ADP
ap-1406	178	7	−	−	PROPN
ap-1406	178	8	1	1	NUM
ap-1406	178	9	.	.	PUNCT
ap-1406	179	1	from	from	ADP
ap-1406	179	2	this	this	PRON
ap-1406	179	3	,	,	PUNCT
ap-1406	179	4	we	we	PRON
ap-1406	179	5	easily	easily	ADV
ap-1406	179	6	derive	derive	VERB
ap-1406	179	7	that	that	SCONJ
ap-1406	179	8	|γ|	|γ|	PROPN
ap-1406	179	9	<	<	X
ap-1406	179	10	2	2	X
ap-1406	179	11	.	.	PUNCT
ap-1406	180	1	as	as	ADP
ap-1406	180	2	a	a	DET
ap-1406	180	3	consequence	consequence	NOUN
ap-1406	180	4	,	,	PUNCT
ap-1406	180	5	we	we	PRON
ap-1406	180	6	can	can	AUX
ap-1406	180	7	easily	easily	ADV
ap-1406	180	8	deduce	deduce	VERB
ap-1406	180	9	the	the	DET
ap-1406	180	10	relation	relation	NOUN
ap-1406	180	11	between	between	ADP
ap-1406	180	12	ito	ito	PROPN
ap-1406	180	13	-	-	PUNCT
ap-1406	180	14	sadahiro	sadahiro	PROPN
ap-1406	180	15	numbers	number	NOUN
ap-1406	180	16	greater	great	ADJ
ap-1406	180	17	or	or	CCONJ
ap-1406	180	18	equal	equal	ADJ
ap-1406	180	19	to	to	ADP
ap-1406	180	20	2	2	NUM
ap-1406	180	21	and	and	CCONJ
ap-1406	180	22	perron	perron	PROPN
ap-1406	180	23	numbers	number	NOUN
ap-1406	180	24	.	.	PUNCT
ap-1406	181	1	corollary	corollary	ADJ
ap-1406	181	2	5	5	NUM
ap-1406	181	3	every	every	DET
ap-1406	181	4	ito	ito	PROPN
ap-1406	181	5	-	-	PROPN
ap-1406	181	6	sadahiro	sadahiro	PROPN
ap-1406	181	7	number	number	NOUN
ap-1406	181	8	β	β	NOUN
ap-1406	181	9	≥	≥	NOUN
ap-1406	181	10	2	2	NUM
ap-1406	181	11	is	be	AUX
ap-1406	181	12	a	a	DET
ap-1406	181	13	perron	perron	PROPN
ap-1406	181	14	number	number	NOUN
ap-1406	181	15	.	.	PUNCT
ap-1406	182	1	in	in	ADP
ap-1406	182	2	a	a	DET
ap-1406	182	3	recent	recent	ADJ
ap-1406	182	4	preprint	preprint	NOUN
ap-1406	182	5	[	[	X
ap-1406	182	6	9	9	NUM
ap-1406	182	7	]	]	PUNCT
ap-1406	182	8	,	,	PUNCT
ap-1406	182	9	it	it	PRON
ap-1406	182	10	is	be	AUX
ap-1406	182	11	shown	show	VERB
ap-1406	182	12	that	that	SCONJ
ap-1406	182	13	also	also	ADV
ap-1406	182	14	itosadahiro	itosadahiro	VERB
ap-1406	182	15	numbers	number	NOUN
ap-1406	182	16	β	β	X
ap-1406	182	17	<	<	X
ap-1406	182	18	2	2	NUM
ap-1406	182	19	are	be	AUX
ap-1406	182	20	perron	perron	PROPN
ap-1406	182	21	numbers	number	NOUN
ap-1406	182	22	.	.	PUNCT
ap-1406	183	1	4	4	NUM
ap-1406	183	2	periodic	periodic	ADJ
ap-1406	183	3	expansions	expansion	NOUN
ap-1406	183	4	in	in	ADP
ap-1406	183	5	the	the	DET
ap-1406	183	6	ito	ito	PROPN
ap-1406	183	7	-	-	PROPN
ap-1406	183	8	sadahiro	sadahiro	PROPN
ap-1406	183	9	system	system	NOUN
ap-1406	183	10	representations	representation	NOUN
ap-1406	183	11	of	of	ADP
ap-1406	183	12	numbers	number	NOUN
ap-1406	183	13	in	in	ADP
ap-1406	183	14	the	the	DET
ap-1406	183	15	numeration	numeration	NOUN
ap-1406	183	16	system	system	NOUN
ap-1406	183	17	with	with	ADP
ap-1406	183	18	a	a	DET
ap-1406	183	19	negative	negative	ADJ
ap-1406	183	20	base	base	NOUN
ap-1406	183	21	from	from	ADP
ap-1406	183	22	the	the	DET
ap-1406	183	23	point	point	NOUN
ap-1406	183	24	of	of	ADP
ap-1406	183	25	view	view	NOUN
ap-1406	183	26	of	of	ADP
ap-1406	183	27	dynamical	dynamical	ADJ
ap-1406	183	28	systems	system	NOUN
ap-1406	183	29	have	have	AUX
ap-1406	183	30	been	be	AUX
ap-1406	183	31	studied	study	VERB
ap-1406	183	32	by	by	ADP
ap-1406	183	33	frougny	frougny	NOUN
ap-1406	183	34	and	and	CCONJ
ap-1406	183	35	lai	lai	X
ap-1406	184	1	[	[	X
ap-1406	184	2	7	7	NUM
ap-1406	184	3	]	]	PUNCT
ap-1406	184	4	.	.	PUNCT
ap-1406	185	1	they	they	PRON
ap-1406	185	2	have	have	AUX
ap-1406	185	3	shown	show	VERB
ap-1406	185	4	the	the	DET
ap-1406	185	5	following	follow	VERB
ap-1406	185	6	statement	statement	NOUN
ap-1406	185	7	.	.	PUNCT
ap-1406	186	1	theorem	theorem	VERB
ap-1406	186	2	6	6	NUM
ap-1406	186	3	if	if	SCONJ
ap-1406	186	4	β	β	NOUN
ap-1406	186	5	is	be	AUX
ap-1406	186	6	a	a	DET
ap-1406	186	7	pisot	pisot	ADJ
ap-1406	186	8	number	number	NOUN
ap-1406	186	9	,	,	PUNCT
ap-1406	186	10	then	then	ADV
ap-1406	186	11	d−β(x	d−β(x	NOUN
ap-1406	186	12	)	)	PUNCT
ap-1406	186	13	is	be	AUX
ap-1406	186	14	eventually	eventually	ADV
ap-1406	186	15	periodic	periodic	ADJ
ap-1406	186	16	for	for	ADP
ap-1406	186	17	any	any	DET
ap-1406	186	18	x	x	SYM
ap-1406	186	19	∈	∈	PROPN
ap-1406	186	20	iβ	iβ	ADP
ap-1406	186	21	∩	∩	NOUN
ap-1406	186	22	q(β	q(β	PRON
ap-1406	186	23	)	)	PUNCT
ap-1406	186	24	.	.	PUNCT
ap-1406	187	1	in	in	ADP
ap-1406	187	2	particular	particular	ADJ
ap-1406	187	3	,	,	PUNCT
ap-1406	187	4	their	their	PRON
ap-1406	187	5	result	result	NOUN
ap-1406	187	6	implies	imply	VERB
ap-1406	187	7	that	that	SCONJ
ap-1406	187	8	every	every	DET
ap-1406	187	9	pisot	pisot	ADJ
ap-1406	187	10	number	number	NOUN
ap-1406	187	11	is	be	AUX
ap-1406	187	12	an	an	DET
ap-1406	187	13	ito	ito	PROPN
ap-1406	187	14	-	-	PUNCT
ap-1406	187	15	sadahiro	sadahiro	NOUN
ap-1406	187	16	number	number	NOUN
ap-1406	187	17	.	.	PUNCT
ap-1406	188	1	here	here	ADV
ap-1406	188	2	,	,	PUNCT
ap-1406	188	3	we	we	PRON
ap-1406	188	4	show	show	VERB
ap-1406	188	5	a	a	DET
ap-1406	188	6	‘	'	PUNCT
ap-1406	188	7	reversed	reversed	ADJ
ap-1406	188	8	’	'	PUNCT
ap-1406	188	9	statement	statement	NOUN
ap-1406	188	10	.	.	PUNCT
ap-1406	189	1	theorem	theorem	VERB
ap-1406	189	2	7	7	NUM
ap-1406	189	3	if	if	SCONJ
ap-1406	189	4	any	any	DET
ap-1406	189	5	x	x	SYM
ap-1406	189	6	∈	∈	NOUN
ap-1406	189	7	iβ	iβ	ADP
ap-1406	189	8	∩q(β	∩q(β	NOUN
ap-1406	189	9	)	)	PUNCT
ap-1406	189	10	has	have	VERB
ap-1406	189	11	eventually	eventually	ADV
ap-1406	189	12	periodic	periodic	ADJ
ap-1406	189	13	(	(	PUNCT
ap-1406	189	14	−β)-expansion	−β)-expansion	NOUN
ap-1406	189	15	,	,	PUNCT
ap-1406	189	16	then	then	ADV
ap-1406	189	17	β	β	PROPN
ap-1406	189	18	is	be	AUX
ap-1406	189	19	either	either	CCONJ
ap-1406	189	20	a	a	DET
ap-1406	189	21	pisot	pisot	ADJ
ap-1406	189	22	number	number	NOUN
ap-1406	189	23	or	or	CCONJ
ap-1406	189	24	a	a	DET
ap-1406	189	25	salem	salem	NOUN
ap-1406	189	26	number	number	NOUN
ap-1406	189	27	.	.	PUNCT
ap-1406	190	1	proof	proof	NOUN
ap-1406	190	2	.	.	PUNCT
ap-1406	191	1	first	first	ADV
ap-1406	191	2	realize	realize	VERB
ap-1406	191	3	that	that	SCONJ
ap-1406	191	4	since	since	SCONJ
ap-1406	191	5	l−β	l−β	NOUN
ap-1406	191	6	∈	∈	PROPN
ap-1406	191	7	q(β	q(β	PROPN
ap-1406	191	8	)	)	PUNCT
ap-1406	191	9	,	,	PUNCT
ap-1406	191	10	by	by	ADP
ap-1406	191	11	assumption	assumption	NOUN
ap-1406	191	12	,	,	PUNCT
ap-1406	191	13	d−β(lβ	d−β(lβ	NOUN
ap-1406	191	14	)	)	PUNCT
ap-1406	191	15	is	be	AUX
ap-1406	191	16	eventually	eventually	ADV
ap-1406	191	17	periodic	periodic	ADJ
ap-1406	191	18	,	,	PUNCT
ap-1406	191	19	and	and	CCONJ
ap-1406	191	20	thus	thus	ADV
ap-1406	191	21	β	β	X
ap-1406	191	22	is	be	AUX
ap-1406	191	23	an	an	DET
ap-1406	191	24	ito	ito	PROPN
ap-1406	191	25	-	-	PUNCT
ap-1406	191	26	sadahiro	sadahiro	NOUN
ap-1406	191	27	number	number	NOUN
ap-1406	191	28	.	.	PUNCT
ap-1406	192	1	therefore	therefore	ADV
ap-1406	192	2	,	,	PUNCT
ap-1406	192	3	using	use	VERB
ap-1406	192	4	corollary	corollary	NOUN
ap-1406	192	5	2	2	NUM
ap-1406	192	6	,	,	PUNCT
ap-1406	192	7	β	β	X
ap-1406	192	8	is	be	AUX
ap-1406	192	9	an	an	DET
ap-1406	192	10	algebraic	algebraic	ADJ
ap-1406	192	11	integer	integer	NOUN
ap-1406	192	12	.	.	PUNCT
ap-1406	193	1	it	it	PRON
ap-1406	193	2	remains	remain	VERB
ap-1406	193	3	to	to	PART
ap-1406	193	4	show	show	VERB
ap-1406	193	5	that	that	SCONJ
ap-1406	193	6	all	all	DET
ap-1406	193	7	conjugates	conjugate	NOUN
ap-1406	193	8	of	of	ADP
ap-1406	193	9	β	β	NOUN
ap-1406	193	10	are	be	AUX
ap-1406	193	11	in	in	ADP
ap-1406	193	12	modulus	modulus	NOUN
ap-1406	193	13	smaller	small	ADJ
ap-1406	193	14	than	than	ADP
ap-1406	193	15	or	or	CCONJ
ap-1406	193	16	equal	equal	ADJ
ap-1406	193	17	to	to	ADP
ap-1406	193	18	1	1	NUM
ap-1406	193	19	.	.	PUNCT
ap-1406	194	1	consider	consider	VERB
ap-1406	194	2	a	a	DET
ap-1406	194	3	real	real	ADJ
ap-1406	194	4	number	number	NOUN
ap-1406	194	5	x	x	PUNCT
ap-1406	194	6	whose	whose	DET
ap-1406	194	7	(	(	PUNCT
ap-1406	194	8	−β)-expansion	−β)-expansion	NOUN
ap-1406	194	9	is	be	AUX
ap-1406	194	10	of	of	ADP
ap-1406	194	11	the	the	DET
ap-1406	194	12	form	form	NOUN
ap-1406	194	13	d−β(x	d−β(x	NOUN
ap-1406	194	14	)	)	PUNCT
ap-1406	194	15	=	=	SYM
ap-1406	194	16	x1x2x3	x1x2x3	PROPN
ap-1406	194	17	.	.	PUNCT
ap-1406	194	18	.	.	PUNCT
ap-1406	195	1	.	.	PUNCT
ap-1406	196	1	we	we	PRON
ap-1406	196	2	now	now	ADV
ap-1406	196	3	show	show	VERB
ap-1406	196	4	that	that	SCONJ
ap-1406	196	5	x1	x1	PROPN
ap-1406	196	6	=	=	PUNCT
ap-1406	196	7	x2	x2	PROPN
ap-1406	196	8	=	=	X
ap-1406	196	9	.	.	PUNCT
ap-1406	196	10	.	.	PUNCT
ap-1406	196	11	.	.	PUNCT
ap-1406	197	1	=	=	PUNCT
ap-1406	197	2	xk−1	xk−1	PROPN
ap-1406	197	3	=	=	SYM
ap-1406	197	4	0	0	NUM
ap-1406	197	5	and	and	CCONJ
ap-1406	197	6	xk	xk	PROPN
ap-1406	197	7	�	�	PROPN
ap-1406	197	8	=	=	SYM
ap-1406	197	9	0	0	NUM
ap-1406	197	10	implies	imply	VERB
ap-1406	197	11	|x|	|x|	PROPN
ap-1406	197	12	≥	≥	NUM
ap-1406	197	13	1	1	NUM
ap-1406	197	14	βk(β	βk(β	NOUN
ap-1406	197	15	+	+	CCONJ
ap-1406	197	16	1	1	NUM
ap-1406	197	17	)	)	PUNCT
ap-1406	197	18	.	.	PUNCT
ap-1406	198	1	(	(	PUNCT
ap-1406	198	2	12	12	NUM
ap-1406	198	3	)	)	PUNCT
ap-1406	198	4	in	in	ADP
ap-1406	198	5	order	order	NOUN
ap-1406	198	6	to	to	PART
ap-1406	198	7	see	see	VERB
ap-1406	198	8	this	this	PRON
ap-1406	198	9	,	,	PUNCT
ap-1406	198	10	we	we	PRON
ap-1406	198	11	estimate	estimate	VERB
ap-1406	198	12	the	the	DET
ap-1406	198	13	series	series	NOUN
ap-1406	198	14	|x|	|x|	PROPN
ap-1406	198	15	=	=	PROPN
ap-1406	198	16	∣∣∣	∣∣∣	PROPN
ap-1406	198	17	xk	xk	PROPN
ap-1406	198	18	(	(	PUNCT
ap-1406	198	19	−β)k	−β)k	NOUN
ap-1406	198	20	+	+	CCONJ
ap-1406	198	21	∞∑	∞∑	NUM
ap-1406	198	22	i=1	i=1	PROPN
ap-1406	198	23	xk+i	xk+i	PROPN
ap-1406	198	24	(	(	PUNCT
ap-1406	198	25	−β)k+i	−β)k+i	NOUN
ap-1406	198	26	∣∣∣	∣∣∣	ADJ
ap-1406	198	27	≥	≥	NUM
ap-1406	198	28	1	1	NUM
ap-1406	198	29	βk	βk	ADP
ap-1406	198	30	−	−	PROPN
ap-1406	198	31	1	1	NUM
ap-1406	198	32	βk	βk	ADP
ap-1406	198	33	∣∣∣	∣∣∣	NOUN
ap-1406	198	34	∞∑	∞∑	NUM
ap-1406	198	35	i=1	i=1	PROPN
ap-1406	198	36	xk+i	xk+i	PROPN
ap-1406	198	37	(	(	PUNCT
ap-1406	198	38	−β)i	−β)i	ADJ
ap-1406	198	39	∣∣∣	∣∣∣	NOUN
ap-1406	198	40	.	.	PUNCT
ap-1406	199	1	since	since	SCONJ
ap-1406	199	2	the	the	DET
ap-1406	199	3	set	set	NOUN
ap-1406	199	4	d−β	d−β	NOUN
ap-1406	199	5	of	of	ADP
ap-1406	199	6	all	all	PRON
ap-1406	199	7	(	(	PUNCT
ap-1406	199	8	−β)-expansions	−β)-expansion	NOUN
ap-1406	199	9	is	be	AUX
ap-1406	199	10	shift	shift	NOUN
ap-1406	199	11	invariant	invariant	ADJ
ap-1406	199	12	,	,	PUNCT
ap-1406	199	13	the	the	DET
ap-1406	199	14	sum	sum	NOUN
ap-1406	199	15	∞∑	∞∑	PROPN
ap-1406	199	16	i=1	i=1	PROPN
ap-1406	199	17	xk+i	xk+i	PROPN
ap-1406	199	18	(	(	PUNCT
ap-1406	199	19	−β)i	−β)i	NOUN
ap-1406	199	20	is	be	AUX
ap-1406	199	21	a	a	DET
ap-1406	199	22	(	(	PUNCT
ap-1406	199	23	−β)-expansion	−β)-expansion	NOUN
ap-1406	199	24	of	of	ADP
ap-1406	199	25	some	some	DET
ap-1406	199	26	y	y	PROPN
ap-1406	199	27	∈	∈	PROPN
ap-1406	199	28	iβ	iβ	ADP
ap-1406	199	29	.	.	PUNCT
ap-1406	200	1	therefore	therefore	ADV
ap-1406	200	2	we	we	PRON
ap-1406	200	3	can	can	AUX
ap-1406	200	4	write	write	VERB
ap-1406	200	5	|x|	|x|	PROPN
ap-1406	200	6	≥	≥	PROPN
ap-1406	200	7	1	1	NUM
ap-1406	200	8	βk	βk	ADP
ap-1406	200	9	−	−	PROPN
ap-1406	200	10	1	1	NUM
ap-1406	200	11	βk	βk	ADP
ap-1406	200	12	|y|	|y|	ADJ
ap-1406	200	13	≥	≥	NOUN
ap-1406	200	14	1	1	NUM
ap-1406	200	15	βk	βk	ADP
ap-1406	200	16	−	−	PROPN
ap-1406	200	17	1	1	NUM
ap-1406	200	18	βk	βk	ADP
ap-1406	200	19	β	β	X
ap-1406	200	20	β	β	X
ap-1406	201	1	+	+	CCONJ
ap-1406	201	2	1	1	NUM
ap-1406	201	3	=	=	SYM
ap-1406	201	4	1	1	NUM
ap-1406	201	5	βk(β	βk(β	NUM
ap-1406	201	6	+	+	NOUN
ap-1406	201	7	1	1	NUM
ap-1406	201	8	)	)	PUNCT
ap-1406	201	9	.	.	PUNCT
ap-1406	202	1	as	as	ADP
ap-1406	202	2	β	β	X
ap-1406	202	3	>	>	X
ap-1406	202	4	1	1	NUM
ap-1406	202	5	,	,	PUNCT
ap-1406	202	6	there	there	PRON
ap-1406	202	7	exists	exist	VERB
ap-1406	202	8	l	l	PROPN
ap-1406	202	9	∈	∈	PROPN
ap-1406	203	1	n	n	PRON
ap-1406	203	2	such	such	ADJ
ap-1406	203	3	that	that	DET
ap-1406	203	4	−	−	PROPN
ap-1406	204	1	β	β	SYM
ap-1406	204	2	β	β	NOUN
ap-1406	205	1	+	+	CCONJ
ap-1406	205	2	1	1	NUM
ap-1406	205	3	<	<	SYM
ap-1406	205	4	1	1	NUM
ap-1406	205	5	(	(	PUNCT
ap-1406	205	6	−β)2l+1	−β)2l+1	X
ap-1406	205	7	.	.	PUNCT
ap-1406	206	1	let	let	VERB
ap-1406	206	2	m	m	PRON
ap-1406	206	3	∈	∈	VERB
ap-1406	206	4	n	n	AUX
ap-1406	206	5	satisfy	satisfy	VERB
ap-1406	206	6	m	m	VERB
ap-1406	206	7	>	>	X
ap-1406	206	8	2l	2l	X
ap-1406	206	9	+	+	CCONJ
ap-1406	206	10	1	1	X
ap-1406	206	11	.	.	X
ap-1406	206	12	choose	choose	VERB
ap-1406	206	13	a	a	DET
ap-1406	206	14	rational	rational	ADJ
ap-1406	206	15	number	number	NOUN
ap-1406	206	16	r	r	NOUN
ap-1406	206	17	such	such	DET
ap-1406	206	18	that	that	DET
ap-1406	206	19	1	1	NUM
ap-1406	206	20	(	(	PUNCT
ap-1406	206	21	−β)2l+1	−β)2l+1	X
ap-1406	206	22	<	<	X
ap-1406	206	23	r	r	X
ap-1406	206	24	<	<	X
ap-1406	206	25	1	1	NUM
ap-1406	206	26	(	(	PUNCT
ap-1406	206	27	−β)2l+1	−β)2l+1	X
ap-1406	206	28	+	+	CCONJ
ap-1406	206	29	1	1	NUM
ap-1406	206	30	βm	βm	NOUN
ap-1406	206	31	(	(	PUNCT
ap-1406	206	32	β	β	X
ap-1406	206	33	+	+	NOUN
ap-1406	206	34	1	1	NUM
ap-1406	206	35	)	)	PUNCT
ap-1406	206	36	.	.	PUNCT
ap-1406	207	1	(	(	PUNCT
ap-1406	207	2	13	13	NUM
ap-1406	207	3	)	)	PUNCT
ap-1406	207	4	according	accord	VERB
ap-1406	207	5	to	to	ADP
ap-1406	207	6	the	the	DET
ap-1406	207	7	auxiliary	auxiliary	ADJ
ap-1406	207	8	statement	statement	NOUN
ap-1406	207	9	(	(	PUNCT
ap-1406	207	10	12	12	NUM
ap-1406	207	11	)	)	PUNCT
ap-1406	207	12	,	,	PUNCT
ap-1406	207	13	the	the	PRON
ap-1406	207	14	(	(	PUNCT
ap-1406	207	15	−β)expansion	−β)expansion	NOUN
ap-1406	207	16	of	of	ADP
ap-1406	207	17	r	r	NOUN
ap-1406	207	18	must	must	AUX
ap-1406	207	19	be	be	AUX
ap-1406	207	20	of	of	ADP
ap-1406	207	21	the	the	DET
ap-1406	207	22	form	form	NOUN
ap-1406	207	23	r	r	NOUN
ap-1406	207	24	=	=	SYM
ap-1406	207	25	1	1	NUM
ap-1406	207	26	(	(	PUNCT
ap-1406	207	27	−β)2l+1	−β)2l+1	X
ap-1406	207	28	+	+	NUM
ap-1406	208	1	∞∑	∞∑	NUM
ap-1406	208	2	i	i	NOUN
ap-1406	208	3	=	=	NOUN
ap-1406	208	4	m+1	m+1	NUM
ap-1406	208	5	ri	ri	PROPN
ap-1406	208	6	(	(	PUNCT
ap-1406	208	7	−β)i	−β)i	NOUN
ap-1406	208	8	.	.	PUNCT
ap-1406	209	1	(	(	PUNCT
ap-1406	209	2	14	14	NUM
ap-1406	209	3	)	)	PUNCT
ap-1406	209	4	62	62	NUM
ap-1406	209	5	acta	acta	PROPN
ap-1406	209	6	polytechnica	polytechnica	PROPN
ap-1406	209	7	vol	vol	NOUN
ap-1406	209	8	.	.	PUNCT
ap-1406	210	1	51	51	NUM
ap-1406	210	2	no	no	INTJ
ap-1406	210	3	.	.	PUNCT
ap-1406	211	1	4/2011	4/2011	NUM
ap-1406	211	2	as	as	SCONJ
ap-1406	211	3	r	r	NOUN
ap-1406	211	4	is	be	AUX
ap-1406	211	5	rational	rational	ADJ
ap-1406	211	6	,	,	PUNCT
ap-1406	211	7	by	by	ADP
ap-1406	211	8	assumption	assumption	NOUN
ap-1406	211	9	,	,	PUNCT
ap-1406	211	10	the	the	DET
ap-1406	211	11	infinite	infinite	ADJ
ap-1406	211	12	word	word	NOUN
ap-1406	211	13	rm+1rm+2	rm+1rm+2	NOUN
ap-1406	211	14	.	.	PUNCT
ap-1406	211	15	.	.	PUNCT
ap-1406	212	1	.	.	PUNCT
ap-1406	213	1	is	be	AUX
ap-1406	213	2	eventually	eventually	ADV
ap-1406	213	3	periodic	periodic	ADJ
ap-1406	213	4	and	and	CCONJ
ap-1406	213	5	by	by	ADP
ap-1406	213	6	summing	sum	VERB
ap-1406	213	7	a	a	DET
ap-1406	213	8	geometric	geometric	ADJ
ap-1406	213	9	series	series	NOUN
ap-1406	213	10	,	,	PUNCT
ap-1406	213	11	the	the	DET
ap-1406	213	12	sum	sum	NOUN
ap-1406	213	13	∞∑	∞∑	PROPN
ap-1406	213	14	i	i	NOUN
ap-1406	213	15	=	=	NOUN
ap-1406	213	16	m+1	m+1	NUM
ap-1406	213	17	ri	ri	PROPN
ap-1406	213	18	(	(	PUNCT
ap-1406	213	19	−β)i	−β)i	NOUN
ap-1406	213	20	can	can	AUX
ap-1406	213	21	be	be	AUX
ap-1406	213	22	rewritten	rewrite	VERB
ap-1406	213	23	as	as	ADP
ap-1406	213	24	∞∑	∞∑	NUM
ap-1406	213	25	i	i	NOUN
ap-1406	213	26	=	=	NOUN
ap-1406	213	27	m+1	m+1	NUM
ap-1406	213	28	ri	ri	PROPN
ap-1406	213	29	(	(	PUNCT
ap-1406	213	30	−β)i	−β)i	NOUN
ap-1406	213	31	=	=	SYM
ap-1406	213	32	c0	c0	NOUN
ap-1406	213	33	+	+	CCONJ
ap-1406	213	34	c1β	c1β	PROPN
ap-1406	213	35	+	+	X
ap-1406	213	36	.	.	PUNCT
ap-1406	213	37	.	.	PUNCT
ap-1406	213	38	.	.	PUNCT
ap-1406	214	1	+	+	CCONJ
ap-1406	214	2	cn−1β	cn−1β	PROPN
ap-1406	214	3	n−1	n−1	PROPN
ap-1406	214	4	∈	∈	PROPN
ap-1406	214	5	q(β	q(β	PROPN
ap-1406	214	6	)	)	PUNCT
ap-1406	214	7	,	,	PUNCT
ap-1406	214	8	where	where	SCONJ
ap-1406	214	9	n	n	PRON
ap-1406	214	10	is	be	AUX
ap-1406	214	11	the	the	DET
ap-1406	214	12	degree	degree	NOUN
ap-1406	214	13	of	of	ADP
ap-1406	214	14	β	β	X
ap-1406	214	15	.	.	PUNCT
ap-1406	215	1	in	in	ADP
ap-1406	215	2	order	order	NOUN
ap-1406	215	3	to	to	PART
ap-1406	215	4	prove	prove	VERB
ap-1406	215	5	the	the	DET
ap-1406	215	6	theorem	theorem	NOUN
ap-1406	215	7	by	by	ADP
ap-1406	215	8	contradiction	contradiction	NOUN
ap-1406	215	9	,	,	PUNCT
ap-1406	215	10	assume	assume	VERB
ap-1406	215	11	that	that	SCONJ
ap-1406	215	12	a	a	DET
ap-1406	215	13	conjugate	conjugate	NOUN
ap-1406	215	14	γ	γ	X
ap-1406	215	15	�	�	PROPN
ap-1406	215	16	=	=	PRON
ap-1406	215	17	β	β	X
ap-1406	215	18	is	be	AUX
ap-1406	215	19	in	in	ADP
ap-1406	215	20	modulus	modulus	NOUN
ap-1406	215	21	greater	great	ADJ
ap-1406	215	22	than	than	ADP
ap-1406	215	23	1	1	NUM
ap-1406	215	24	.	.	PUNCT
ap-1406	215	25	by	by	ADP
ap-1406	215	26	application	application	NOUN
ap-1406	215	27	of	of	ADP
ap-1406	215	28	the	the	DET
ap-1406	215	29	isomorphism	isomorphism	NOUN
ap-1406	215	30	between	between	ADP
ap-1406	215	31	q(β	q(β	PROPN
ap-1406	215	32	)	)	PUNCT
ap-1406	215	33	and	and	CCONJ
ap-1406	215	34	q(γ	q(γ	PROPN
ap-1406	215	35	)	)	PUNCT
ap-1406	215	36	,	,	PUNCT
ap-1406	215	37	we	we	PRON
ap-1406	215	38	get	get	VERB
ap-1406	215	39	c0	c0	NOUN
ap-1406	215	40	+	+	X
ap-1406	215	41	c1γ	c1γ	PROPN
ap-1406	215	42	+	+	CCONJ
ap-1406	215	43	.	.	PUNCT
ap-1406	215	44	.	.	PUNCT
ap-1406	215	45	.	.	PUNCT
ap-1406	216	1	+	+	PUNCT
ap-1406	216	2	cn−1γ	cn−1γ	NOUN
ap-1406	216	3	n−1	n−1	PROPN
ap-1406	216	4	=	=	PUNCT
ap-1406	217	1	∞∑	∞∑	NUM
ap-1406	217	2	i	i	PRON
ap-1406	217	3	=	=	NOUN
ap-1406	217	4	m+1	m+1	NUM
ap-1406	217	5	ri	ri	PROPN
ap-1406	217	6	(	(	PUNCT
ap-1406	217	7	−γ)i	−γ)i	NOUN
ap-1406	217	8	,	,	PUNCT
ap-1406	217	9	and	and	CCONJ
ap-1406	217	10	thus	thus	ADV
ap-1406	217	11	r	r	NOUN
ap-1406	217	12	=	=	SYM
ap-1406	217	13	1	1	NUM
ap-1406	217	14	(	(	PUNCT
ap-1406	217	15	−γ)2l+1	−γ)2l+1	NOUN
ap-1406	217	16	+	+	NUM
ap-1406	217	17	∞∑	∞∑	NUM
ap-1406	217	18	i	i	NOUN
ap-1406	217	19	=	=	NOUN
ap-1406	217	20	m+1	m+1	NUM
ap-1406	217	21	ri	ri	PROPN
ap-1406	217	22	(	(	PUNCT
ap-1406	217	23	−γ)i	−γ)i	NOUN
ap-1406	217	24	.	.	PUNCT
ap-1406	218	1	(	(	PUNCT
ap-1406	218	2	15	15	X
ap-1406	218	3	)	)	PUNCT
ap-1406	218	4	subtracting	subtract	VERB
ap-1406	218	5	(	(	PUNCT
ap-1406	218	6	15	15	NUM
ap-1406	218	7	)	)	PUNCT
ap-1406	218	8	from	from	ADP
ap-1406	218	9	(	(	PUNCT
ap-1406	218	10	14	14	NUM
ap-1406	218	11	)	)	PUNCT
ap-1406	218	12	,	,	PUNCT
ap-1406	218	13	we	we	PRON
ap-1406	218	14	obtain	obtain	VERB
ap-1406	218	15	0	0	NUM
ap-1406	218	16	<	<	X
ap-1406	218	17	∣∣∣	∣∣∣	ADJ
ap-1406	218	18	1	1	NUM
ap-1406	218	19	(	(	PUNCT
ap-1406	218	20	−β)2l+1	−β)2l+1	X
ap-1406	218	21	−	−	PROPN
ap-1406	218	22	1	1	NUM
ap-1406	218	23	(	(	PUNCT
ap-1406	218	24	−γ)2l+1	−γ)2l+1	NOUN
ap-1406	218	25	∣∣∣	∣∣∣	NOUN
ap-1406	218	26	≤	≤	PROPN
ap-1406	218	27	(	(	PUNCT
ap-1406	218	28	16	16	NUM
ap-1406	218	29	)	)	PUNCT
ap-1406	219	1	∞∑	∞∑	NUM
ap-1406	219	2	i	i	PRON
ap-1406	219	3	=	=	NOUN
ap-1406	219	4	m+1	m+1	NUM
ap-1406	219	5	ri	ri	NOUN
ap-1406	219	6	∣∣(−β)−i	∣∣(−β)−i	ADV
ap-1406	219	7	−	−	PROPN
ap-1406	219	8	(	(	PUNCT
ap-1406	219	9	−γ)−i	−γ)−i	NOUN
ap-1406	219	10	∣∣	∣∣	NUM
ap-1406	219	11	≤	≤	X
ap-1406	219	12	2#β$ηm+1	2#β$ηm+1	NUM
ap-1406	219	13	1	1	NUM
ap-1406	219	14	−	−	PROPN
ap-1406	219	15	η	η	PROPN
ap-1406	219	16	,	,	PUNCT
ap-1406	219	17	where	where	SCONJ
ap-1406	219	18	η	η	PROPN
ap-1406	219	19	=	=	SYM
ap-1406	219	20	max{|β|−1	max{|β|−1	PROPN
ap-1406	219	21	,	,	PUNCT
ap-1406	219	22	|γ|−1	|γ|−1	NUM
ap-1406	219	23	}	}	PUNCT
ap-1406	219	24	<	<	X
ap-1406	219	25	1	1	X
ap-1406	219	26	.	.	PUNCT
ap-1406	220	1	obviously	obviously	ADV
ap-1406	220	2	,	,	PUNCT
ap-1406	220	3	for	for	ADP
ap-1406	220	4	any	any	DET
ap-1406	220	5	m	m	NOUN
ap-1406	220	6	>	>	X
ap-1406	220	7	2l	2l	NOUN
ap-1406	220	8	+	+	CCONJ
ap-1406	220	9	1	1	NUM
ap-1406	220	10	,	,	PUNCT
ap-1406	220	11	we	we	PRON
ap-1406	220	12	can	can	AUX
ap-1406	220	13	find	find	VERB
ap-1406	220	14	a	a	DET
ap-1406	220	15	rational	rational	ADJ
ap-1406	220	16	r	r	NOUN
ap-1406	220	17	satisfying	satisfying	NOUN
ap-1406	220	18	(	(	PUNCT
ap-1406	220	19	13	13	NUM
ap-1406	220	20	)	)	PUNCT
ap-1406	220	21	and	and	CCONJ
ap-1406	220	22	thus	thus	ADV
ap-1406	220	23	derive	derive	VERB
ap-1406	220	24	the	the	DET
ap-1406	220	25	inequality	inequality	NOUN
ap-1406	220	26	(	(	PUNCT
ap-1406	220	27	16	16	NUM
ap-1406	220	28	)	)	PUNCT
ap-1406	220	29	.	.	PUNCT
ap-1406	221	1	however	however	ADV
ap-1406	221	2	,	,	PUNCT
ap-1406	221	3	the	the	DET
ap-1406	221	4	left	leave	VERB
ap-1406	221	5	-	-	PUNCT
ap-1406	221	6	hand	hand	NOUN
ap-1406	221	7	side	side	NOUN
ap-1406	221	8	of	of	ADP
ap-1406	221	9	(	(	PUNCT
ap-1406	221	10	16	16	NUM
ap-1406	221	11	)	)	PUNCT
ap-1406	221	12	is	be	AUX
ap-1406	221	13	a	a	DET
ap-1406	221	14	fixed	fix	VERB
ap-1406	221	15	positive	positive	ADJ
ap-1406	221	16	number	number	NOUN
ap-1406	221	17	,	,	PUNCT
ap-1406	221	18	whereas	whereas	SCONJ
ap-1406	221	19	the	the	DET
ap-1406	221	20	right	right	ADJ
ap-1406	221	21	-	-	PUNCT
ap-1406	221	22	hand	hand	NOUN
ap-1406	221	23	side	side	NOUN
ap-1406	221	24	decreases	decrease	VERB
ap-1406	221	25	to	to	ADP
ap-1406	221	26	zero	zero	NUM
ap-1406	221	27	with	with	ADP
ap-1406	221	28	increasing	increase	VERB
ap-1406	221	29	m	m	VERB
ap-1406	221	30	,	,	PUNCT
ap-1406	221	31	which	which	PRON
ap-1406	221	32	is	be	AUX
ap-1406	221	33	a	a	DET
ap-1406	221	34	contradiction	contradiction	NOUN
ap-1406	221	35	.	.	PUNCT
ap-1406	222	1	in	in	ADP
ap-1406	222	2	order	order	NOUN
ap-1406	222	3	to	to	PART
ap-1406	222	4	stress	stress	VERB
ap-1406	222	5	the	the	DET
ap-1406	222	6	analogy	analogy	NOUN
ap-1406	222	7	of	of	ADP
ap-1406	222	8	the	the	DET
ap-1406	222	9	ito	ito	PROPN
ap-1406	222	10	-	-	PROPN
ap-1406	222	11	sadahiro	sadahiro	PROPN
ap-1406	222	12	numeration	numeration	NOUN
ap-1406	222	13	system	system	NOUN
ap-1406	222	14	with	with	ADP
ap-1406	222	15	rényi	rényi	PROPN
ap-1406	222	16	β	β	NOUN
ap-1406	222	17	-	-	NOUN
ap-1406	222	18	expansions	expansion	NOUN
ap-1406	222	19	of	of	ADP
ap-1406	222	20	numbers	number	NOUN
ap-1406	222	21	,	,	PUNCT
ap-1406	222	22	recall	recall	VERB
ap-1406	222	23	that	that	SCONJ
ap-1406	222	24	already	already	ADV
ap-1406	222	25	schmidt	schmidt	VERB
ap-1406	222	26	in	in	ADP
ap-1406	222	27	[	[	X
ap-1406	222	28	13	13	NUM
ap-1406	222	29	]	]	PUNCT
ap-1406	222	30	has	have	AUX
ap-1406	222	31	shown	show	VERB
ap-1406	222	32	that	that	SCONJ
ap-1406	222	33	for	for	ADP
ap-1406	222	34	a	a	DET
ap-1406	222	35	pisot	pisot	ADJ
ap-1406	222	36	number	number	NOUN
ap-1406	222	37	β	β	NOUN
ap-1406	222	38	,	,	PUNCT
ap-1406	222	39	any	any	DET
ap-1406	222	40	x	x	SYM
ap-1406	222	41	∈	∈	PROPN
ap-1406	223	1	[	[	X
ap-1406	223	2	0	0	NUM
ap-1406	223	3	,	,	PUNCT
ap-1406	223	4	1	1	NUM
ap-1406	223	5	)	)	PUNCT
ap-1406	223	6	∩	∩	NOUN
ap-1406	223	7	q(β	q(β	NOUN
ap-1406	223	8	)	)	PUNCT
ap-1406	223	9	has	have	VERB
ap-1406	223	10	an	an	DET
ap-1406	223	11	eventually	eventually	ADV
ap-1406	223	12	periodic	periodic	ADJ
ap-1406	223	13	β	β	NOUN
ap-1406	223	14	-	-	NOUN
ap-1406	223	15	expansion	expansion	NOUN
ap-1406	223	16	and	and	CCONJ
ap-1406	223	17	also	also	ADV
ap-1406	223	18	,	,	PUNCT
ap-1406	223	19	conversely	conversely	ADV
ap-1406	223	20	,	,	PUNCT
ap-1406	223	21	that	that	SCONJ
ap-1406	223	22	every	every	DET
ap-1406	223	23	x	x	X
ap-1406	223	24	∈	∈	PROPN
ap-1406	223	25	[	[	X
ap-1406	223	26	0	0	NUM
ap-1406	223	27	,	,	PUNCT
ap-1406	223	28	1	1	NUM
ap-1406	223	29	)	)	PUNCT
ap-1406	223	30	∩	∩	NOUN
ap-1406	223	31	q(β	q(β	NOUN
ap-1406	223	32	)	)	PUNCT
ap-1406	223	33	having	have	VERB
ap-1406	223	34	an	an	DET
ap-1406	223	35	eventually	eventually	ADV
ap-1406	223	36	periodic	periodic	ADJ
ap-1406	223	37	β	β	ADJ
ap-1406	223	38	-	-	ADJ
ap-1406	223	39	expansion	expansion	NOUN
ap-1406	223	40	force	force	NOUN
ap-1406	223	41	β	β	PROPN
ap-1406	223	42	is	be	AUX
ap-1406	223	43	either	either	CCONJ
ap-1406	223	44	a	a	DET
ap-1406	223	45	pisot	pisot	ADJ
ap-1406	223	46	number	number	NOUN
ap-1406	223	47	or	or	CCONJ
ap-1406	223	48	a	a	DET
ap-1406	223	49	salem	salem	NOUN
ap-1406	223	50	number	number	NOUN
ap-1406	223	51	.	.	PUNCT
ap-1406	224	1	in	in	ADP
ap-1406	224	2	fact	fact	NOUN
ap-1406	224	3	,	,	PUNCT
ap-1406	224	4	the	the	DET
ap-1406	224	5	proof	proof	NOUN
ap-1406	224	6	of	of	ADP
ap-1406	224	7	theorem	theorem	NOUN
ap-1406	224	8	6	6	NUM
ap-1406	224	9	given	give	VERB
ap-1406	224	10	by	by	ADP
ap-1406	224	11	frougny	frougny	NOUN
ap-1406	224	12	and	and	CCONJ
ap-1406	224	13	lai	lai	PROPN
ap-1406	224	14	,	,	PUNCT
ap-1406	224	15	as	as	ADV
ap-1406	224	16	well	well	ADV
ap-1406	224	17	as	as	ADP
ap-1406	224	18	our	our	PRON
ap-1406	224	19	proof	proof	NOUN
ap-1406	224	20	of	of	ADP
ap-1406	224	21	theorem	theorem	ADJ
ap-1406	224	22	7	7	NUM
ap-1406	224	23	are	be	AUX
ap-1406	224	24	using	use	VERB
ap-1406	224	25	the	the	DET
ap-1406	224	26	ideas	idea	NOUN
ap-1406	224	27	presented	present	VERB
ap-1406	224	28	in	in	ADP
ap-1406	224	29	[	[	X
ap-1406	224	30	13	13	NUM
ap-1406	224	31	]	]	PUNCT
ap-1406	224	32	.	.	PUNCT
ap-1406	225	1	a	a	DET
ap-1406	225	2	special	special	ADJ
ap-1406	225	3	case	case	NOUN
ap-1406	225	4	of	of	ADP
ap-1406	225	5	numbers	number	NOUN
ap-1406	225	6	with	with	ADP
ap-1406	225	7	periodic	periodic	ADJ
ap-1406	225	8	(	(	PUNCT
ap-1406	225	9	−β)expansion	−β)expansion	PROPN
ap-1406	225	10	is	be	AUX
ap-1406	225	11	given	give	VERB
ap-1406	225	12	by	by	ADP
ap-1406	225	13	those	those	DET
ap-1406	225	14	numbers	number	NOUN
ap-1406	225	15	x	x	PUNCT
ap-1406	225	16	for	for	ADP
ap-1406	225	17	which	which	PRON
ap-1406	225	18	the	the	DET
ap-1406	225	19	infinite	infinite	ADJ
ap-1406	225	20	word	word	NOUN
ap-1406	225	21	d−β(x	d−β(x	NOUN
ap-1406	225	22	)	)	PUNCT
ap-1406	225	23	has	have	VERB
ap-1406	225	24	suffix	suffix	ADJ
ap-1406	225	25	0ω	0ω	NOUN
ap-1406	225	26	.	.	PUNCT
ap-1406	226	1	we	we	PRON
ap-1406	226	2	then	then	ADV
ap-1406	226	3	say	say	VERB
ap-1406	226	4	that	that	SCONJ
ap-1406	226	5	the	the	DET
ap-1406	226	6	expansion	expansion	NOUN
ap-1406	226	7	d−β(x	d−β(x	NOUN
ap-1406	226	8	)	)	PUNCT
ap-1406	226	9	is	be	AUX
ap-1406	226	10	finite	finite	PROPN
ap-1406	226	11	.	.	PUNCT
ap-1406	227	1	an	an	DET
ap-1406	227	2	example	example	NOUN
ap-1406	227	3	of	of	ADP
ap-1406	227	4	such	such	DET
ap-1406	227	5	a	a	DET
ap-1406	227	6	number	number	NOUN
ap-1406	227	7	is	be	AUX
ap-1406	227	8	x	x	X
ap-1406	227	9	=	=	SYM
ap-1406	227	10	0	0	NUM
ap-1406	227	11	with	with	ADP
ap-1406	227	12	(	(	PUNCT
ap-1406	227	13	−β)-expansion	−β)-expansion	NOUN
ap-1406	227	14	d−β(x	d−β(x	NOUN
ap-1406	227	15	)	)	PUNCT
ap-1406	227	16	=	=	SYM
ap-1406	227	17	0ω	0ω	NOUN
ap-1406	227	18	.	.	PUNCT
ap-1406	228	1	as	as	SCONJ
ap-1406	228	2	is	be	AUX
ap-1406	228	3	shown	show	VERB
ap-1406	228	4	in	in	ADP
ap-1406	228	5	[	[	X
ap-1406	228	6	10	10	NUM
ap-1406	228	7	]	]	PUNCT
ap-1406	228	8	,	,	PUNCT
ap-1406	228	9	if	if	SCONJ
ap-1406	228	10	β	β	X
ap-1406	228	11	<	<	X
ap-1406	228	12	1	1	NUM
ap-1406	228	13	2	2	NUM
ap-1406	228	14	(	(	PUNCT
ap-1406	228	15	1	1	NUM
ap-1406	228	16	+	+	CCONJ
ap-1406	228	17	√	√	NUM
ap-1406	228	18	5	5	NUM
ap-1406	228	19	)	)	PUNCT
ap-1406	228	20	,	,	PUNCT
ap-1406	228	21	then	then	ADV
ap-1406	228	22	x	x	X
ap-1406	228	23	=	=	SYM
ap-1406	228	24	0	0	NUM
ap-1406	228	25	is	be	AUX
ap-1406	228	26	the	the	DET
ap-1406	228	27	only	only	ADJ
ap-1406	228	28	number	number	NOUN
ap-1406	228	29	with	with	ADP
ap-1406	228	30	finite	finite	NOUN
ap-1406	228	31	(	(	PUNCT
ap-1406	228	32	−β)-expansion	−β)-expansion	NOUN
ap-1406	228	33	.	.	PUNCT
ap-1406	229	1	this	this	DET
ap-1406	229	2	property	property	NOUN
ap-1406	229	3	of	of	ADP
ap-1406	229	4	the	the	DET
ap-1406	229	5	ito	ito	PROPN
ap-1406	229	6	-	-	PROPN
ap-1406	229	7	sadahiro	sadahiro	PROPN
ap-1406	229	8	numeration	numeration	NOUN
ap-1406	229	9	system	system	NOUN
ap-1406	229	10	has	have	VERB
ap-1406	229	11	no	no	DET
ap-1406	229	12	analogue	analogue	NOUN
ap-1406	229	13	in	in	ADP
ap-1406	229	14	rényi	rényi	PROPN
ap-1406	229	15	β	β	NOUN
ap-1406	229	16	-	-	NOUN
ap-1406	229	17	expansions	expansion	NOUN
ap-1406	229	18	;	;	PUNCT
ap-1406	229	19	for	for	ADP
ap-1406	229	20	positive	positive	ADJ
ap-1406	229	21	base	base	NOUN
ap-1406	229	22	,	,	PUNCT
ap-1406	229	23	the	the	DET
ap-1406	229	24	set	set	NOUN
ap-1406	229	25	of	of	ADP
ap-1406	229	26	finite	finite	ADJ
ap-1406	229	27	β	β	NOUN
ap-1406	229	28	-	-	NOUN
ap-1406	229	29	expansions	expansion	NOUN
ap-1406	229	30	is	be	AUX
ap-1406	229	31	always	always	ADV
ap-1406	229	32	dense	dense	ADJ
ap-1406	229	33	in	in	ADP
ap-1406	229	34	[	[	X
ap-1406	229	35	0	0	NUM
ap-1406	229	36	,	,	PUNCT
ap-1406	229	37	1	1	NUM
ap-1406	229	38	)	)	PUNCT
ap-1406	229	39	.	.	PUNCT
ap-1406	230	1	just	just	ADV
ap-1406	230	2	as	as	SCONJ
ap-1406	230	3	in	in	ADP
ap-1406	230	4	the	the	DET
ap-1406	230	5	numeration	numeration	NOUN
ap-1406	230	6	system	system	NOUN
ap-1406	230	7	with	with	ADP
ap-1406	230	8	a	a	DET
ap-1406	230	9	positive	positive	ADJ
ap-1406	230	10	base	base	NOUN
ap-1406	230	11	,	,	PUNCT
ap-1406	230	12	we	we	PRON
ap-1406	230	13	can	can	AUX
ap-1406	230	14	extend	extend	VERB
ap-1406	230	15	the	the	DET
ap-1406	230	16	definition	definition	NOUN
ap-1406	230	17	of	of	ADP
ap-1406	230	18	(	(	PUNCT
ap-1406	230	19	−β)-expansions	−β)-expansion	NOUN
ap-1406	230	20	of	of	ADP
ap-1406	230	21	x	x	INTJ
ap-1406	230	22	to	to	ADP
ap-1406	230	23	all	all	DET
ap-1406	230	24	real	real	ADJ
ap-1406	230	25	numbers	number	NOUN
ap-1406	230	26	x	x	ADP
ap-1406	230	27	,	,	PUNCT
ap-1406	230	28	and	and	CCONJ
ap-1406	230	29	define	define	VERB
ap-1406	230	30	the	the	DET
ap-1406	230	31	notion	notion	NOUN
ap-1406	230	32	of	of	ADP
ap-1406	230	33	a	a	DET
ap-1406	230	34	(	(	PUNCT
ap-1406	230	35	−β)-integer	−β)-integer	NOUN
ap-1406	230	36	as	as	ADP
ap-1406	230	37	a	a	DET
ap-1406	230	38	real	real	ADJ
ap-1406	230	39	number	number	NOUN
ap-1406	230	40	y	y	PRON
ap-1406	230	41	such	such	ADJ
ap-1406	230	42	that	that	PRON
ap-1406	230	43	y	y	PROPN
ap-1406	230	44	=	=	PUNCT
ap-1406	230	45	yk(−β)k	yk(−β)k	NOUN
ap-1406	230	46	+	+	CCONJ
ap-1406	230	47	.	.	PUNCT
ap-1406	230	48	.	.	PUNCT
ap-1406	230	49	.	.	PUNCT
ap-1406	231	1	+	+	CCONJ
ap-1406	231	2	y1(−β	y1(−β	NUM
ap-1406	231	3	)	)	PUNCT
ap-1406	232	1	+	+	CCONJ
ap-1406	232	2	y0	y0	NOUN
ap-1406	232	3	,	,	PUNCT
ap-1406	232	4	where	where	SCONJ
ap-1406	232	5	yk	yk	PROPN
ap-1406	232	6	.	.	PUNCT
ap-1406	232	7	.	.	PUNCT
ap-1406	232	8	.	.	PUNCT
ap-1406	233	1	y1y00ω	y1y00ω	NOUN
ap-1406	233	2	is	be	AUX
ap-1406	233	3	the	the	DET
ap-1406	233	4	(	(	PUNCT
ap-1406	233	5	−β)-expansion	−β)-expansion	NOUN
ap-1406	233	6	of	of	ADP
ap-1406	233	7	some	some	DET
ap-1406	233	8	number	number	NOUN
ap-1406	233	9	in	in	ADP
ap-1406	233	10	iβ	iβ	ADP
ap-1406	233	11	.	.	PUNCT
ap-1406	234	1	the	the	DET
ap-1406	234	2	set	set	NOUN
ap-1406	234	3	of	of	ADP
ap-1406	234	4	(	(	PUNCT
ap-1406	234	5	−β)-integers	−β)-integers	PROPN
ap-1406	234	6	is	be	AUX
ap-1406	234	7	denoted	denote	VERB
ap-1406	234	8	by	by	ADP
ap-1406	234	9	z−β	z−β	PROPN
ap-1406	234	10	.	.	PUNCT
ap-1406	235	1	with	with	ADP
ap-1406	235	2	this	this	DET
ap-1406	235	3	notation	notation	NOUN
ap-1406	235	4	,	,	PUNCT
ap-1406	235	5	we	we	PRON
ap-1406	235	6	can	can	AUX
ap-1406	235	7	write	write	VERB
ap-1406	235	8	the	the	DET
ap-1406	235	9	set	set	NOUN
ap-1406	235	10	of	of	ADP
ap-1406	235	11	all	all	DET
ap-1406	235	12	numbers	number	NOUN
ap-1406	235	13	with	with	ADP
ap-1406	235	14	finite	finite	NOUN
ap-1406	235	15	(	(	PUNCT
ap-1406	235	16	−β)-expansions	−β)-expansion	NOUN
ap-1406	235	17	as	as	ADP
ap-1406	235	18	fin(−β	fin(−β	PROPN
ap-1406	235	19	)	)	PUNCT
ap-1406	235	20	=	=	SYM
ap-1406	235	21	∞⋃	∞⋃	PROPN
ap-1406	235	22	k=0	k=0	PROPN
ap-1406	235	23	1	1	NUM
ap-1406	235	24	(	(	PUNCT
ap-1406	235	25	−β)k	−β)k	ADV
ap-1406	235	26	z−β	z−β	NUM
ap-1406	235	27	.	.	PUNCT
ap-1406	236	1	it	it	PRON
ap-1406	236	2	is	be	AUX
ap-1406	236	3	not	not	PART
ap-1406	236	4	surprising	surprising	ADJ
ap-1406	236	5	that	that	SCONJ
ap-1406	236	6	the	the	DET
ap-1406	236	7	arithmetical	arithmetical	ADJ
ap-1406	236	8	properties	property	NOUN
ap-1406	236	9	of	of	ADP
ap-1406	236	10	β	β	NOUN
ap-1406	236	11	-	-	NOUN
ap-1406	236	12	expansions	expansion	NOUN
ap-1406	236	13	and	and	CCONJ
ap-1406	236	14	(	(	PUNCT
ap-1406	236	15	−β)-expansions	−β)-expansion	NOUN
ap-1406	236	16	depend	depend	VERB
ap-1406	236	17	on	on	ADP
ap-1406	236	18	the	the	DET
ap-1406	236	19	choice	choice	NOUN
ap-1406	236	20	of	of	ADP
ap-1406	236	21	the	the	DET
ap-1406	236	22	base	base	NOUN
ap-1406	236	23	β	β	NOUN
ap-1406	236	24	.	.	PUNCT
ap-1406	237	1	it	it	PRON
ap-1406	237	2	can	can	AUX
ap-1406	237	3	be	be	AUX
ap-1406	237	4	shown	show	VERB
ap-1406	237	5	that	that	SCONJ
ap-1406	237	6	both	both	CCONJ
ap-1406	237	7	zβ	zβ	PROPN
ap-1406	237	8	and	and	CCONJ
ap-1406	237	9	z−β	z−β	PROPN
ap-1406	237	10	is	be	AUX
ap-1406	237	11	closed	close	VERB
ap-1406	237	12	under	under	ADP
ap-1406	237	13	addition	addition	NOUN
ap-1406	237	14	and	and	CCONJ
ap-1406	237	15	multiplication	multiplication	NOUN
ap-1406	237	16	if	if	SCONJ
ap-1406	237	17	and	and	CCONJ
ap-1406	237	18	only	only	ADV
ap-1406	237	19	if	if	SCONJ
ap-1406	237	20	β	β	X
ap-1406	237	21	∈	∈	PROPN
ap-1406	237	22	n.	n.	NOUN
ap-1406	237	23	on	on	ADP
ap-1406	237	24	the	the	DET
ap-1406	237	25	other	other	ADJ
ap-1406	237	26	hand	hand	NOUN
ap-1406	237	27	,	,	PUNCT
ap-1406	237	28	fin(β	fin(β	PROPN
ap-1406	237	29	)	)	PUNCT
ap-1406	237	30	and	and	CCONJ
ap-1406	237	31	fin(−β	fin(−β	PROPN
ap-1406	237	32	)	)	PUNCT
ap-1406	237	33	can	can	AUX
ap-1406	237	34	have	have	VERB
ap-1406	237	35	a	a	DET
ap-1406	237	36	ring	ring	NOUN
ap-1406	237	37	structure	structure	NOUN
ap-1406	237	38	even	even	ADV
ap-1406	237	39	if	if	SCONJ
ap-1406	237	40	β	β	NOUN
ap-1406	237	41	is	be	AUX
ap-1406	237	42	not	not	PART
ap-1406	237	43	an	an	DET
ap-1406	237	44	integer	integer	NOUN
ap-1406	237	45	.	.	PUNCT
ap-1406	238	1	frougny	frougny	NOUN
ap-1406	238	2	and	and	CCONJ
ap-1406	238	3	solomyak	solomyak	NOUN
ap-1406	239	1	[	[	X
ap-1406	239	2	8	8	NUM
ap-1406	239	3	]	]	PUNCT
ap-1406	239	4	have	have	AUX
ap-1406	239	5	shown	show	VERB
ap-1406	239	6	that	that	SCONJ
ap-1406	239	7	if	if	SCONJ
ap-1406	239	8	fin(β	fin(β	PROPN
ap-1406	239	9	)	)	PUNCT
ap-1406	239	10	is	be	AUX
ap-1406	239	11	a	a	DET
ap-1406	239	12	ring	ring	NOUN
ap-1406	239	13	,	,	PUNCT
ap-1406	239	14	then	then	ADV
ap-1406	239	15	β	β	X
ap-1406	239	16	is	be	AUX
ap-1406	239	17	a	a	DET
ap-1406	239	18	pisot	pisot	ADJ
ap-1406	239	19	number	number	NOUN
ap-1406	239	20	.	.	PUNCT
ap-1406	240	1	a	a	DET
ap-1406	240	2	similar	similar	ADJ
ap-1406	240	3	result	result	NOUN
ap-1406	240	4	is	be	AUX
ap-1406	240	5	given	give	VERB
ap-1406	240	6	in	in	ADP
ap-1406	240	7	[	[	X
ap-1406	240	8	10	10	NUM
ap-1406	240	9	]	]	PUNCT
ap-1406	240	10	for	for	ADP
ap-1406	240	11	a	a	DET
ap-1406	240	12	negative	negative	ADJ
ap-1406	240	13	base	base	NOUN
ap-1406	240	14	:	:	PUNCT
ap-1406	240	15	fin(−β	fin(−β	PROPN
ap-1406	240	16	)	)	PUNCT
ap-1406	240	17	being	be	AUX
ap-1406	240	18	a	a	DET
ap-1406	240	19	ring	ring	NOUN
ap-1406	240	20	implies	imply	VERB
ap-1406	240	21	that	that	SCONJ
ap-1406	240	22	β	β	NOUN
ap-1406	240	23	is	be	AUX
ap-1406	240	24	either	either	CCONJ
ap-1406	240	25	a	a	DET
ap-1406	240	26	pisot	pisot	ADJ
ap-1406	240	27	number	number	NOUN
ap-1406	240	28	or	or	CCONJ
ap-1406	240	29	a	a	DET
ap-1406	240	30	salem	salem	NOUN
ap-1406	240	31	number	number	NOUN
ap-1406	240	32	.	.	PUNCT
ap-1406	241	1	in	in	ADP
ap-1406	241	2	[	[	X
ap-1406	241	3	10	10	NUM
ap-1406	241	4	]	]	PUNCT
ap-1406	241	5	we	we	PRON
ap-1406	241	6	also	also	ADV
ap-1406	241	7	prove	prove	VERB
ap-1406	241	8	the	the	DET
ap-1406	241	9	conjecture	conjecture	NOUN
ap-1406	241	10	of	of	ADP
ap-1406	241	11	ito	ito	PROPN
ap-1406	241	12	and	and	CCONJ
ap-1406	241	13	sadahiro	sadahiro	VERB
ap-1406	241	14	that	that	SCONJ
ap-1406	241	15	in	in	ADP
ap-1406	241	16	the	the	DET
ap-1406	241	17	case	case	NOUN
ap-1406	241	18	of	of	ADP
ap-1406	241	19	quadratic	quadratic	ADJ
ap-1406	241	20	pisot	pisot	ADJ
ap-1406	241	21	base	base	NOUN
ap-1406	241	22	β	β	X
ap-1406	241	23	the	the	DET
ap-1406	241	24	set	set	NOUN
ap-1406	241	25	fin(−β	fin(−β	PROPN
ap-1406	241	26	)	)	PUNCT
ap-1406	241	27	is	be	AUX
ap-1406	241	28	a	a	DET
ap-1406	241	29	ring	ring	NOUN
ap-1406	241	30	if	if	SCONJ
ap-1406	241	31	and	and	CCONJ
ap-1406	241	32	only	only	ADV
ap-1406	241	33	if	if	SCONJ
ap-1406	241	34	the	the	DET
ap-1406	241	35	conjugate	conjugate	NOUN
ap-1406	241	36	of	of	ADP
ap-1406	241	37	β	β	PROPN
ap-1406	241	38	is	be	AUX
ap-1406	241	39	negative	negative	ADJ
ap-1406	241	40	.	.	PUNCT
ap-1406	242	1	5	5	NUM
ap-1406	242	2	comments	comment	NOUN
ap-1406	242	3	and	and	CCONJ
ap-1406	242	4	open	open	ADJ
ap-1406	242	5	questions	question	NOUN
ap-1406	242	6	•	•	NOUN
ap-1406	242	7	every	every	DET
ap-1406	242	8	pisot	pisot	ADJ
ap-1406	242	9	number	number	NOUN
ap-1406	242	10	is	be	AUX
ap-1406	242	11	a	a	DET
ap-1406	242	12	parry	parry	NOUN
ap-1406	242	13	number	number	NOUN
ap-1406	242	14	and	and	CCONJ
ap-1406	242	15	every	every	DET
ap-1406	242	16	parry	parry	NOUN
ap-1406	242	17	number	number	NOUN
ap-1406	242	18	is	be	AUX
ap-1406	242	19	a	a	DET
ap-1406	242	20	perron	perron	PROPN
ap-1406	242	21	number	number	NOUN
ap-1406	242	22	,	,	PUNCT
ap-1406	242	23	and	and	CCONJ
ap-1406	242	24	neither	neither	PRON
ap-1406	242	25	of	of	ADP
ap-1406	242	26	these	these	DET
ap-1406	242	27	statements	statement	NOUN
ap-1406	242	28	can	can	AUX
ap-1406	242	29	be	be	AUX
ap-1406	242	30	reversed	reverse	VERB
ap-1406	242	31	.	.	PUNCT
ap-1406	243	1	the	the	DET
ap-1406	243	2	former	former	ADJ
ap-1406	243	3	is	be	AUX
ap-1406	243	4	a	a	DET
ap-1406	243	5	consequence	consequence	NOUN
ap-1406	243	6	of	of	ADP
ap-1406	243	7	the	the	DET
ap-1406	243	8	mentioned	mention	VERB
ap-1406	243	9	result	result	NOUN
ap-1406	243	10	of	of	ADP
ap-1406	243	11	schmidt	schmidt	NOUN
ap-1406	243	12	,	,	PUNCT
ap-1406	243	13	the	the	DET
ap-1406	243	14	latter	latter	ADJ
ap-1406	243	15	statement	statement	NOUN
ap-1406	243	16	follows	follow	VERB
ap-1406	243	17	for	for	ADP
ap-1406	243	18	example	example	NOUN
ap-1406	243	19	from	from	ADP
ap-1406	243	20	the	the	DET
ap-1406	243	21	fact	fact	NOUN
ap-1406	243	22	that	that	SCONJ
ap-1406	243	23	every	every	DET
ap-1406	243	24	perron	perron	PROPN
ap-1406	243	25	number	number	NOUN
ap-1406	243	26	has	have	AUX
ap-1406	243	27	an	an	DET
ap-1406	243	28	associated	associated	ADJ
ap-1406	243	29	canonical	canonical	ADJ
ap-1406	243	30	substitution	substitution	NOUN
ap-1406	243	31	ϕβ	ϕβ	NOUN
ap-1406	243	32	,	,	PUNCT
ap-1406	243	33	see	see	VERB
ap-1406	243	34	[	[	X
ap-1406	243	35	4	4	NUM
ap-1406	243	36	]	]	PUNCT
ap-1406	243	37	.	.	PUNCT
ap-1406	244	1	the	the	DET
ap-1406	244	2	substitution	substitution	NOUN
ap-1406	244	3	is	be	AUX
ap-1406	244	4	primitive	primitive	ADJ
ap-1406	244	5	,	,	PUNCT
ap-1406	244	6	and	and	CCONJ
ap-1406	244	7	its	its	PRON
ap-1406	244	8	incidence	incidence	NOUN
ap-1406	244	9	matrix	matrix	NOUN
ap-1406	244	10	has	have	VERB
ap-1406	244	11	β	β	NOUN
ap-1406	244	12	as	as	ADP
ap-1406	244	13	its	its	PRON
ap-1406	244	14	eigenvalue	eigenvalue	NOUN
ap-1406	244	15	.	.	PUNCT
ap-1406	245	1	the	the	DET
ap-1406	245	2	fixed	fixed	ADJ
ap-1406	245	3	point	point	NOUN
ap-1406	245	4	of	of	ADP
ap-1406	245	5	ϕβ	ϕβ	PROPN
ap-1406	245	6	is	be	AUX
ap-1406	245	7	an	an	DET
ap-1406	245	8	infinite	infinite	ADJ
ap-1406	245	9	word	word	NOUN
ap-1406	245	10	which	which	PRON
ap-1406	245	11	codes	code	VERB
ap-1406	245	12	the	the	DET
ap-1406	245	13	sequence	sequence	NOUN
ap-1406	245	14	of	of	ADP
ap-1406	245	15	distances	distance	NOUN
ap-1406	245	16	between	between	ADP
ap-1406	245	17	consecutive	consecutive	ADJ
ap-1406	245	18	β	β	NOUN
ap-1406	245	19	-	-	NOUN
ap-1406	245	20	integers	integer	NOUN
ap-1406	245	21	.	.	PUNCT
ap-1406	246	1	•	•	NUM
ap-1406	246	2	for	for	ADP
ap-1406	246	3	the	the	DET
ap-1406	246	4	negative	negative	ADJ
ap-1406	246	5	base	base	NOUN
ap-1406	246	6	numeration	numeration	NOUN
ap-1406	246	7	system	system	NOUN
ap-1406	246	8	,	,	PUNCT
ap-1406	246	9	we	we	PRON
ap-1406	246	10	can	can	AUX
ap-1406	246	11	derive	derive	VERB
ap-1406	246	12	from	from	ADP
ap-1406	246	13	theorem	theorem	NOUN
ap-1406	246	14	6	6	NUM
ap-1406	246	15	that	that	SCONJ
ap-1406	246	16	every	every	DET
ap-1406	246	17	pisot	pisot	ADJ
ap-1406	246	18	number	number	NOUN
ap-1406	246	19	is	be	AUX
ap-1406	246	20	an	an	DET
ap-1406	246	21	ito	ito	PROPN
ap-1406	246	22	-	-	PUNCT
ap-1406	246	23	sadahiro	sadahiro	NOUN
ap-1406	246	24	number	number	NOUN
ap-1406	246	25	.	.	PUNCT
ap-1406	247	1	from	from	ADP
ap-1406	247	2	corollary	corollary	ADJ
ap-1406	247	3	5	5	NUM
ap-1406	247	4	we	we	PRON
ap-1406	247	5	know	know	VERB
ap-1406	247	6	that	that	SCONJ
ap-1406	247	7	an	an	DET
ap-1406	247	8	ito	ito	PROPN
ap-1406	247	9	-	-	PUNCT
ap-1406	247	10	sadahiro	sadahiro	PROPN
ap-1406	247	11	number	number	NOUN
ap-1406	247	12	β	β	NOUN
ap-1406	247	13	≥	≥	NOUN
ap-1406	247	14	2	2	NUM
ap-1406	247	15	is	be	AUX
ap-1406	247	16	a	a	DET
ap-1406	247	17	perron	perron	PROPN
ap-1406	247	18	number	number	NOUN
ap-1406	247	19	.	.	PUNCT
ap-1406	248	1	based	base	VERB
ap-1406	248	2	on	on	ADP
ap-1406	248	3	our	our	PRON
ap-1406	248	4	investigation	investigation	NOUN
ap-1406	248	5	,	,	PUNCT
ap-1406	248	6	we	we	PRON
ap-1406	248	7	conjecture	conjecture	VERB
ap-1406	248	8	that	that	SCONJ
ap-1406	248	9	for	for	ADP
ap-1406	248	10	any	any	DET
ap-1406	248	11	ito	ito	PROPN
ap-1406	248	12	-	-	PUNCT
ap-1406	248	13	sadahiro	sadahiro	PROPN
ap-1406	248	14	number	number	NOUN
ap-1406	248	15	β	β	NOUN
ap-1406	248	16	≥	≥	NOUN
ap-1406	248	17	1	1	NUM
ap-1406	248	18	2	2	NUM
ap-1406	248	19	(	(	PUNCT
ap-1406	248	20	1	1	NUM
ap-1406	248	21	+	+	CCONJ
ap-1406	248	22	√	√	NUM
ap-1406	248	23	5	5	NUM
ap-1406	248	24	)	)	PUNCT
ap-1406	248	25	,	,	PUNCT
ap-1406	248	26	the	the	DET
ap-1406	248	27	sequence	sequence	NOUN
ap-1406	248	28	of	of	ADP
ap-1406	248	29	distances	distance	NOUN
ap-1406	248	30	between	between	ADP
ap-1406	248	31	consecutive	consecutive	ADJ
ap-1406	248	32	(	(	PUNCT
ap-1406	248	33	−β)-integers	−β)-integer	NOUN
ap-1406	248	34	can	can	AUX
ap-1406	248	35	be	be	AUX
ap-1406	248	36	coded	code	VERB
ap-1406	248	37	by	by	ADP
ap-1406	248	38	a	a	DET
ap-1406	248	39	fixed	fix	VERB
ap-1406	248	40	point	point	NOUN
ap-1406	248	41	of	of	ADP
ap-1406	248	42	a	a	DET
ap-1406	248	43	‘	'	PUNCT
ap-1406	248	44	canonical	canonical	ADJ
ap-1406	248	45	’	'	PUNCT
ap-1406	248	46	substitution	substitution	NOUN
ap-1406	248	47	which	which	PRON
ap-1406	248	48	is	be	AUX
ap-1406	248	49	primitive	primitive	ADJ
ap-1406	248	50	and	and	CCONJ
ap-1406	248	51	its	its	PRON
ap-1406	248	52	incidence	incidence	NOUN
ap-1406	248	53	matrix	matrix	NOUN
ap-1406	248	54	has	have	AUX
ap-1406	248	55	β2	β2	VERB
ap-1406	248	56	for	for	ADP
ap-1406	248	57	its	its	PRON
ap-1406	248	58	dominant	dominant	ADJ
ap-1406	248	59	eigenvalue	eigenvalue	NOUN
ap-1406	248	60	.	.	PUNCT
ap-1406	249	1	thus	thus	ADV
ap-1406	249	2	we	we	PRON
ap-1406	249	3	expect	expect	VERB
ap-1406	249	4	that	that	SCONJ
ap-1406	249	5	63	63	NUM
ap-1406	249	6	acta	acta	PROPN
ap-1406	249	7	polytechnica	polytechnica	PROPN
ap-1406	249	8	vol	vol	NOUN
ap-1406	249	9	.	.	PUNCT
ap-1406	250	1	51	51	NUM
ap-1406	250	2	no	no	INTJ
ap-1406	250	3	.	.	PUNCT
ap-1406	251	1	4/2011	4/2011	NUM
ap-1406	252	1	every	every	DET
ap-1406	252	2	ito	ito	PROPN
ap-1406	252	3	-	-	PROPN
ap-1406	252	4	sadahiro	sadahiro	PROPN
ap-1406	252	5	number	number	NOUN
ap-1406	252	6	β	β	NOUN
ap-1406	252	7	≥	≥	NOUN
ap-1406	252	8	1	1	NUM
ap-1406	252	9	2	2	NUM
ap-1406	252	10	(	(	PUNCT
ap-1406	252	11	1	1	NUM
ap-1406	252	12	+	+	CCONJ
ap-1406	252	13	√	√	NUM
ap-1406	252	14	5	5	NUM
ap-1406	252	15	)	)	PUNCT
ap-1406	252	16	is	be	AUX
ap-1406	252	17	also	also	ADV
ap-1406	252	18	a	a	DET
ap-1406	252	19	perron	perron	PROPN
ap-1406	252	20	number	number	NOUN
ap-1406	252	21	.	.	PUNCT
ap-1406	253	1	in	in	ADP
ap-1406	253	2	the	the	DET
ap-1406	253	3	case	case	NOUN
ap-1406	253	4	that	that	SCONJ
ap-1406	253	5	β	β	NOUN
ap-1406	253	6	<	<	X
ap-1406	253	7	1	1	NUM
ap-1406	253	8	2	2	NUM
ap-1406	253	9	(	(	PUNCT
ap-1406	253	10	1	1	NUM
ap-1406	253	11	+	+	CCONJ
ap-1406	253	12	√	√	NUM
ap-1406	253	13	5	5	NUM
ap-1406	253	14	)	)	PUNCT
ap-1406	253	15	,	,	PUNCT
ap-1406	253	16	we	we	PRON
ap-1406	253	17	have	have	AUX
ap-1406	253	18	z−β	z−β	NUM
ap-1406	253	19	=	=	SYM
ap-1406	253	20	{	{	PUNCT
ap-1406	253	21	0	0	NUM
ap-1406	253	22	}	}	PUNCT
ap-1406	253	23	and	and	CCONJ
ap-1406	253	24	so	so	ADV
ap-1406	253	25	the	the	DET
ap-1406	253	26	situation	situation	NOUN
ap-1406	253	27	is	be	AUX
ap-1406	253	28	not	not	PART
ap-1406	253	29	at	at	ADV
ap-1406	253	30	all	all	ADV
ap-1406	253	31	obvious	obvious	ADJ
ap-1406	253	32	.	.	PUNCT
ap-1406	254	1	•	•	NOUN
ap-1406	254	2	in	in	ADP
ap-1406	254	3	[	[	X
ap-1406	254	4	14	14	NUM
ap-1406	254	5	]	]	PUNCT
ap-1406	254	6	,	,	PUNCT
ap-1406	254	7	solomyak	solomyak	NOUN
ap-1406	254	8	has	have	AUX
ap-1406	254	9	explicitly	explicitly	ADV
ap-1406	254	10	described	describe	VERB
ap-1406	254	11	the	the	DET
ap-1406	254	12	set	set	NOUN
ap-1406	254	13	of	of	ADP
ap-1406	254	14	conjugates	conjugate	NOUN
ap-1406	254	15	of	of	ADP
ap-1406	254	16	all	all	DET
ap-1406	254	17	parry	parry	NOUN
ap-1406	254	18	numbers	number	NOUN
ap-1406	254	19	.	.	PUNCT
ap-1406	255	1	in	in	ADP
ap-1406	255	2	particular	particular	ADJ
ap-1406	255	3	,	,	PUNCT
ap-1406	255	4	he	he	PRON
ap-1406	255	5	has	have	AUX
ap-1406	255	6	shown	show	VERB
ap-1406	255	7	that	that	SCONJ
ap-1406	255	8	this	this	DET
ap-1406	255	9	set	set	NOUN
ap-1406	255	10	is	be	AUX
ap-1406	255	11	included	include	VERB
ap-1406	255	12	in	in	ADP
ap-1406	255	13	the	the	DET
ap-1406	255	14	complex	complex	ADJ
ap-1406	255	15	disc	disc	NOUN
ap-1406	255	16	of	of	ADP
ap-1406	255	17	radius	radius	NOUN
ap-1406	255	18	1	1	NUM
ap-1406	255	19	2	2	NUM
ap-1406	255	20	(	(	PUNCT
ap-1406	255	21	1	1	NUM
ap-1406	255	22	+	+	CCONJ
ap-1406	255	23	√	√	NUM
ap-1406	255	24	5	5	NUM
ap-1406	255	25	)	)	PUNCT
ap-1406	255	26	,	,	PUNCT
ap-1406	255	27	and	and	CCONJ
ap-1406	255	28	that	that	SCONJ
ap-1406	255	29	this	this	DET
ap-1406	255	30	radius	radius	NOUN
ap-1406	255	31	can	can	AUX
ap-1406	255	32	not	not	PART
ap-1406	255	33	be	be	AUX
ap-1406	255	34	diminished	diminish	VERB
ap-1406	255	35	.	.	PUNCT
ap-1406	256	1	for	for	ADP
ap-1406	256	2	his	his	PRON
ap-1406	256	3	proof	proof	NOUN
ap-1406	256	4	it	it	PRON
ap-1406	256	5	was	be	AUX
ap-1406	256	6	important	important	ADJ
ap-1406	256	7	that	that	SCONJ
ap-1406	256	8	all	all	DET
ap-1406	256	9	conjugates	conjugate	NOUN
ap-1406	256	10	of	of	ADP
ap-1406	256	11	a	a	DET
ap-1406	256	12	parry	parry	NOUN
ap-1406	256	13	number	number	NOUN
ap-1406	256	14	are	be	AUX
ap-1406	256	15	roots	root	NOUN
ap-1406	256	16	of	of	ADP
ap-1406	256	17	a	a	DET
ap-1406	256	18	polynomial	polynomial	NOUN
ap-1406	256	19	with	with	ADP
ap-1406	256	20	real	real	ADJ
ap-1406	256	21	coefficients	coefficient	NOUN
ap-1406	256	22	in	in	ADP
ap-1406	256	23	the	the	DET
ap-1406	256	24	interval	interval	NOUN
ap-1406	256	25	[	[	X
ap-1406	256	26	0	0	NUM
ap-1406	256	27	,	,	PUNCT
ap-1406	256	28	1	1	NUM
ap-1406	256	29	)	)	PUNCT
ap-1406	256	30	.	.	PUNCT
ap-1406	257	1	in	in	ADP
ap-1406	257	2	the	the	DET
ap-1406	257	3	proof	proof	NOUN
ap-1406	257	4	of	of	ADP
ap-1406	257	5	theorem	theorem	NOUN
ap-1406	257	6	4	4	NUM
ap-1406	257	7	we	we	PRON
ap-1406	257	8	show	show	VERB
ap-1406	257	9	that	that	SCONJ
ap-1406	257	10	conjugates	conjugate	NOUN
ap-1406	257	11	of	of	ADP
ap-1406	257	12	an	an	DET
ap-1406	257	13	ito	ito	PROPN
ap-1406	257	14	-	-	PUNCT
ap-1406	257	15	sadahiro	sadahiro	NOUN
ap-1406	257	16	number	number	NOUN
ap-1406	257	17	are	be	AUX
ap-1406	257	18	roots	root	NOUN
ap-1406	257	19	of	of	ADP
ap-1406	257	20	a	a	DET
ap-1406	257	21	polynomial	polynomial	ADJ
ap-1406	257	22	(	(	PUNCT
ap-1406	257	23	11	11	NUM
ap-1406	257	24	)	)	PUNCT
ap-1406	257	25	with	with	ADP
ap-1406	257	26	coefficients	coefficient	NOUN
ap-1406	257	27	in	in	ADP
ap-1406	257	28	[	[	X
ap-1406	257	29	−1	−1	NOUN
ap-1406	257	30	,	,	PUNCT
ap-1406	257	31	1	1	NUM
ap-1406	257	32	]	]	PUNCT
ap-1406	257	33	.	.	PUNCT
ap-1406	258	1	from	from	ADP
ap-1406	258	2	this	this	PRON
ap-1406	258	3	,	,	PUNCT
ap-1406	258	4	we	we	PRON
ap-1406	258	5	derive	derive	VERB
ap-1406	258	6	that	that	SCONJ
ap-1406	258	7	conjugates	conjugate	NOUN
ap-1406	258	8	of	of	ADP
ap-1406	258	9	ito	ito	PROPN
ap-1406	258	10	-	-	PUNCT
ap-1406	258	11	sadahiro	sadahiro	PROPN
ap-1406	258	12	numbers	number	NOUN
ap-1406	258	13	lie	lie	VERB
ap-1406	258	14	in	in	ADP
ap-1406	258	15	the	the	DET
ap-1406	258	16	complex	complex	ADJ
ap-1406	258	17	disc	disc	NOUN
ap-1406	258	18	of	of	ADP
ap-1406	258	19	radius	radius	NOUN
ap-1406	258	20	≤	≤	NOUN
ap-1406	258	21	2	2	NUM
ap-1406	258	22	.	.	PUNCT
ap-1406	259	1	we	we	PRON
ap-1406	259	2	do	do	AUX
ap-1406	259	3	not	not	PART
ap-1406	259	4	know	know	VERB
ap-1406	259	5	whether	whether	SCONJ
ap-1406	259	6	this	this	DET
ap-1406	259	7	value	value	NOUN
ap-1406	259	8	can	can	AUX
ap-1406	259	9	be	be	AUX
ap-1406	259	10	diminished	diminish	VERB
ap-1406	259	11	.	.	PUNCT
ap-1406	260	1	acknowledgement	acknowledgement	NOUN
ap-1406	260	2	we	we	PRON
ap-1406	260	3	acknowledge	acknowledge	VERB
ap-1406	260	4	financial	financial	ADJ
ap-1406	260	5	support	support	NOUN
ap-1406	260	6	from	from	ADP
ap-1406	260	7	czech	czech	PROPN
ap-1406	260	8	science	science	NOUN
ap-1406	260	9	foundation	foundation	NOUN
ap-1406	260	10	grant	grant	VERB
ap-1406	260	11	201/09/0584	201/09/0584	NUM
ap-1406	260	12	and	and	CCONJ
ap-1406	260	13	from	from	ADP
ap-1406	260	14	grants	grant	NOUN
ap-1406	260	15	msm6840770039	msm6840770039	NOUN
ap-1406	260	16	and	and	CCONJ
ap-1406	260	17	lc06002	lc06002	NOUN
ap-1406	260	18	of	of	ADP
ap-1406	260	19	the	the	DET
ap-1406	260	20	ministry	ministry	PROPN
ap-1406	260	21	of	of	ADP
ap-1406	260	22	education	education	PROPN
ap-1406	260	23	,	,	PUNCT
ap-1406	260	24	youth	youth	NOUN
ap-1406	260	25	,	,	PUNCT
ap-1406	260	26	and	and	CCONJ
ap-1406	260	27	sports	sport	NOUN
ap-1406	260	28	of	of	ADP
ap-1406	260	29	the	the	DET
ap-1406	260	30	czech	czech	PROPN
ap-1406	260	31	republic	republic	NOUN
ap-1406	260	32	.	.	PUNCT
ap-1406	261	1	references	reference	NOUN
ap-1406	261	2	[	[	X
ap-1406	261	3	1	1	NUM
ap-1406	261	4	]	]	PUNCT
ap-1406	261	5	ambrož	ambrož	ADV
ap-1406	261	6	,	,	PUNCT
ap-1406	261	7	p.	p.	NOUN
ap-1406	261	8	,	,	PUNCT
ap-1406	261	9	dombek	dombek	PROPN
ap-1406	261	10	,	,	PUNCT
ap-1406	261	11	d.	d.	PROPN
ap-1406	261	12	,	,	PUNCT
ap-1406	261	13	masáková	masáková	PROPN
ap-1406	261	14	,	,	PUNCT
ap-1406	261	15	z.	z.	PROPN
ap-1406	261	16	,	,	PUNCT
ap-1406	261	17	pelantová	pelantová	PROPN
ap-1406	261	18	,	,	PUNCT
ap-1406	261	19	e.	e.	PROPN
ap-1406	261	20	:	:	PUNCT
ap-1406	261	21	numbers	number	NOUN
ap-1406	261	22	with	with	ADP
ap-1406	261	23	integer	integer	NOUN
ap-1406	261	24	expansion	expansion	NOUN
ap-1406	261	25	in	in	ADP
ap-1406	261	26	the	the	DET
ap-1406	261	27	numeration	numeration	NOUN
ap-1406	261	28	system	system	NOUN
ap-1406	261	29	with	with	ADP
ap-1406	261	30	negative	negative	ADJ
ap-1406	261	31	base	base	NOUN
ap-1406	261	32	,	,	PUNCT
ap-1406	261	33	preprint	preprint	NOUN
ap-1406	261	34	2009	2009	NUM
ap-1406	261	35	,	,	PUNCT
ap-1406	261	36	13pp	13pp	NOUN
ap-1406	261	37	.	.	PUNCT
ap-1406	261	38	http://arxiv.org/abs/0912.4597	http://arxiv.org/abs/0912.4597	PROPN
ap-1406	262	1	[	[	X
ap-1406	262	2	2	2	NUM
ap-1406	262	3	]	]	PUNCT
ap-1406	262	4	bassino	bassino	NOUN
ap-1406	262	5	,	,	PUNCT
ap-1406	262	6	f.	f.	PROPN
ap-1406	262	7	:	:	PUNCT
ap-1406	262	8	β	β	NOUN
ap-1406	262	9	-	-	NOUN
ap-1406	262	10	expansions	expansion	NOUN
ap-1406	262	11	for	for	ADP
ap-1406	262	12	cubic	cubic	ADJ
ap-1406	262	13	pisot	pisot	ADJ
ap-1406	262	14	numbers	number	NOUN
ap-1406	262	15	,	,	PUNCT
ap-1406	262	16	5th	5th	ADJ
ap-1406	262	17	latin	latin	ADJ
ap-1406	262	18	american	american	ADJ
ap-1406	262	19	theoretical	theoretical	ADJ
ap-1406	262	20	informatics	informatics	PROPN
ap-1406	262	21	symposium	symposium	NOUN
ap-1406	262	22	(	(	PUNCT
ap-1406	262	23	latin’02	latin’02	PROPN
ap-1406	262	24	)	)	PUNCT
ap-1406	262	25	,	,	PUNCT
ap-1406	262	26	2286	2286	NUM
ap-1406	262	27	lncs	lnc	NOUN
ap-1406	262	28	.	.	PUNCT
ap-1406	263	1	cancun	cancun	PROPN
ap-1406	263	2	,	,	PUNCT
ap-1406	263	3	mexico	mexico	PROPN
ap-1406	263	4	.	.	PUNCT
ap-1406	264	1	april	april	PROPN
ap-1406	264	2	,	,	PUNCT
ap-1406	264	3	2002	2002	NUM
ap-1406	264	4	.	.	PUNCT
ap-1406	265	1	pp	pp	ADJ
ap-1406	265	2	.	.	PUNCT
ap-1406	266	1	141–152	141–152	NUM
ap-1406	266	2	.	.	PUNCT
ap-1406	266	3	springerverlag	springerverlag	NOUN
ap-1406	266	4	.	.	PUNCT
ap-1406	267	1	[	[	X
ap-1406	267	2	3	3	X
ap-1406	267	3	]	]	SYM
ap-1406	267	4	burd́ık	burd́ık	PROPN
ap-1406	267	5	,	,	PUNCT
ap-1406	267	6	č.	č.	PROPN
ap-1406	267	7	,	,	PUNCT
ap-1406	267	8	frougny	frougny	NOUN
ap-1406	267	9	,	,	PUNCT
ap-1406	267	10	ch	ch	NOUN
ap-1406	267	11	.	.	PROPN
ap-1406	267	12	,	,	PUNCT
ap-1406	267	13	gazeau	gazeau	PROPN
ap-1406	267	14	,	,	PUNCT
ap-1406	267	15	j.	j.	PROPN
ap-1406	267	16	p.	p.	PROPN
ap-1406	267	17	,	,	PUNCT
ap-1406	267	18	krejcar	krejcar	PROPN
ap-1406	267	19	,	,	PUNCT
ap-1406	267	20	r.	r.	PROPN
ap-1406	267	21	:	:	PUNCT
ap-1406	267	22	beta	beta	NOUN
ap-1406	267	23	-	-	PUNCT
ap-1406	267	24	integers	integer	NOUN
ap-1406	267	25	as	as	ADP
ap-1406	267	26	natural	natural	ADJ
ap-1406	267	27	counting	counting	NOUN
ap-1406	267	28	systems	system	NOUN
ap-1406	267	29	for	for	ADP
ap-1406	267	30	quasicrystals	quasicrystal	NOUN
ap-1406	267	31	,	,	PUNCT
ap-1406	267	32	j.	j.	PROPN
ap-1406	267	33	phys	phys	PROPN
ap-1406	267	34	.	.	PUNCT
ap-1406	268	1	a	a	DET
ap-1406	268	2	:	:	PUNCT
ap-1406	268	3	math	math	NOUN
ap-1406	268	4	.	.	PUNCT
ap-1406	269	1	gen	gen	PROPN
ap-1406	269	2	.	.	PROPN
ap-1406	269	3	31	31	NUM
ap-1406	269	4	(	(	PUNCT
ap-1406	269	5	1998	1998	NUM
ap-1406	269	6	)	)	PUNCT
ap-1406	269	7	,	,	PUNCT
ap-1406	269	8	6	6	NUM
ap-1406	269	9	449–6472	449–6472	NOUN
ap-1406	269	10	.	.	PUNCT
ap-1406	270	1	[	[	X
ap-1406	270	2	4	4	NUM
ap-1406	270	3	]	]	X
ap-1406	270	4	fabre	fabre	PROPN
ap-1406	270	5	,	,	PUNCT
ap-1406	270	6	s.	s.	PROPN
ap-1406	270	7	:	:	PUNCT
ap-1406	270	8	substitutions	substitution	NOUN
ap-1406	270	9	et	et	NOUN
ap-1406	270	10	β	β	X
ap-1406	270	11	-	-	PUNCT
ap-1406	270	12	systèmes	systèmes	PROPN
ap-1406	270	13	de	de	X
ap-1406	270	14	numération	numération	PROPN
ap-1406	270	15	,	,	PUNCT
ap-1406	270	16	theoret	theoret	ADJ
ap-1406	270	17	.	.	PUNCT
ap-1406	271	1	comput	comput	NOUN
ap-1406	271	2	.	.	PUNCT
ap-1406	272	1	sci	sci	PROPN
ap-1406	272	2	.	.	PROPN
ap-1406	272	3	137	137	NUM
ap-1406	272	4	(	(	PUNCT
ap-1406	272	5	1995	1995	NUM
ap-1406	272	6	)	)	PUNCT
ap-1406	272	7	,	,	PUNCT
ap-1406	272	8	219–236	219–236	NUM
ap-1406	272	9	.	.	PUNCT
ap-1406	273	1	[	[	X
ap-1406	273	2	5	5	NUM
ap-1406	273	3	]	]	SYM
ap-1406	273	4	ito	ito	PROPN
ap-1406	273	5	,	,	PUNCT
ap-1406	273	6	s.	s.	PROPN
ap-1406	273	7	,	,	PUNCT
ap-1406	273	8	sadahiro	sadahiro	PROPN
ap-1406	273	9	,	,	PUNCT
ap-1406	273	10	t.	t.	PROPN
ap-1406	273	11	:	:	PUNCT
ap-1406	273	12	(	(	PUNCT
ap-1406	273	13	−β)-expansions	−β)-expansion	NOUN
ap-1406	273	14	of	of	ADP
ap-1406	273	15	real	real	ADJ
ap-1406	273	16	numbers	number	NOUN
ap-1406	273	17	,	,	PUNCT
ap-1406	273	18	integers	integer	NOUN
ap-1406	273	19	9	9	NUM
ap-1406	273	20	(	(	PUNCT
ap-1406	273	21	2009	2009	NUM
ap-1406	273	22	)	)	PUNCT
ap-1406	273	23	,	,	PUNCT
ap-1406	273	24	239–259	239–259	NUM
ap-1406	273	25	.	.	PUNCT
ap-1406	274	1	[	[	X
ap-1406	274	2	6	6	NUM
ap-1406	274	3	]	]	SYM
ap-1406	274	4	ito	ito	PROPN
ap-1406	274	5	,	,	PUNCT
ap-1406	274	6	s.	s.	PROPN
ap-1406	274	7	,	,	PUNCT
ap-1406	274	8	takahashi	takahashi	PROPN
ap-1406	274	9	,	,	PUNCT
ap-1406	274	10	y.	y.	PROPN
ap-1406	274	11	:	:	PUNCT
ap-1406	274	12	markov	markov	PROPN
ap-1406	274	13	subshifts	subshift	NOUN
ap-1406	274	14	and	and	CCONJ
ap-1406	274	15	realization	realization	NOUN
ap-1406	274	16	of	of	ADP
ap-1406	274	17	β	β	NOUN
ap-1406	274	18	-	-	NOUN
ap-1406	274	19	expansions	expansion	NOUN
ap-1406	274	20	,	,	PUNCT
ap-1406	274	21	j.	j.	PROPN
ap-1406	274	22	math	math	PROPN
ap-1406	274	23	.	.	PUNCT
ap-1406	275	1	soc	soc	PROPN
ap-1406	275	2	.	.	PUNCT
ap-1406	276	1	japan	japan	PROPN
ap-1406	276	2	26	26	NUM
ap-1406	276	3	(	(	PUNCT
ap-1406	276	4	1974	1974	NUM
ap-1406	276	5	)	)	PUNCT
ap-1406	276	6	,	,	PUNCT
ap-1406	276	7	33–55	33–55	NUM
ap-1406	276	8	.	.	PUNCT
ap-1406	277	1	[	[	X
ap-1406	277	2	7	7	NUM
ap-1406	277	3	]	]	SYM
ap-1406	277	4	frougny	frougny	NOUN
ap-1406	277	5	,	,	PUNCT
ap-1406	277	6	ch	ch	NOUN
ap-1406	277	7	.	.	PROPN
ap-1406	277	8	,	,	PUNCT
ap-1406	277	9	lai	lai	PROPN
ap-1406	277	10	,	,	PUNCT
ap-1406	277	11	a.	a.	PROPN
ap-1406	277	12	c.	c.	PROPN
ap-1406	277	13	:	:	PUNCT
ap-1406	277	14	negative	negative	ADJ
ap-1406	277	15	bases	basis	NOUN
ap-1406	277	16	and	and	CCONJ
ap-1406	277	17	automata	automata	NOUN
ap-1406	277	18	,	,	PUNCT
ap-1406	277	19	discr	discr	PROPN
ap-1406	277	20	.	.	PUNCT
ap-1406	277	21	math	math	NOUN
ap-1406	277	22	.	.	PUNCT
ap-1406	278	1	theor	theor	PROPN
ap-1406	278	2	.	.	PUNCT
ap-1406	279	1	comp	comp	PROPN
ap-1406	279	2	.	.	PUNCT
ap-1406	280	1	sci	sci	PROPN
ap-1406	280	2	.	.	PROPN
ap-1406	280	3	13	13	NUM
ap-1406	280	4	,	,	PUNCT
ap-1406	280	5	no	no	DET
ap-1406	280	6	1	1	NUM
ap-1406	280	7	(	(	PUNCT
ap-1406	280	8	2011	2011	NUM
ap-1406	280	9	)	)	PUNCT
ap-1406	280	10	,	,	PUNCT
ap-1406	280	11	75–94	75–94	X
ap-1406	280	12	.	.	PUNCT
ap-1406	281	1	[	[	X
ap-1406	281	2	8	8	NUM
ap-1406	281	3	]	]	SYM
ap-1406	281	4	frougny	frougny	NOUN
ap-1406	281	5	,	,	PUNCT
ap-1406	281	6	ch	ch	NOUN
ap-1406	281	7	.	.	PROPN
ap-1406	281	8	,	,	PUNCT
ap-1406	281	9	solomyak	solomyak	PROPN
ap-1406	281	10	,	,	PUNCT
ap-1406	281	11	b.	b.	PROPN
ap-1406	281	12	:	:	PUNCT
ap-1406	281	13	finite	finite	VERB
ap-1406	281	14	β	β	NOUN
ap-1406	281	15	-	-	NOUN
ap-1406	281	16	expansions	expansion	NOUN
ap-1406	281	17	,	,	PUNCT
ap-1406	281	18	ergodic	ergodic	ADJ
ap-1406	281	19	theory	theory	NOUN
ap-1406	281	20	dynamical	dynamical	ADJ
ap-1406	281	21	systems	system	NOUN
ap-1406	281	22	12	12	NUM
ap-1406	281	23	(	(	PUNCT
ap-1406	281	24	1994	1994	NUM
ap-1406	281	25	)	)	PUNCT
ap-1406	281	26	,	,	PUNCT
ap-1406	281	27	713–723	713–723	NUM
ap-1406	281	28	.	.	PUNCT
ap-1406	282	1	[	[	X
ap-1406	282	2	9	9	NUM
ap-1406	282	3	]	]	X
ap-1406	282	4	liao	liao	PROPN
ap-1406	282	5	,	,	PUNCT
ap-1406	282	6	l.	l.	PROPN
ap-1406	282	7	,	,	PUNCT
ap-1406	282	8	steiner	steiner	PROPN
ap-1406	282	9	,	,	PUNCT
ap-1406	282	10	w.	w.	NOUN
ap-1406	282	11	:	:	PUNCT
ap-1406	282	12	dynamical	dynamical	ADJ
ap-1406	282	13	properties	property	NOUN
ap-1406	282	14	of	of	ADP
ap-1406	282	15	the	the	DET
ap-1406	282	16	negative	negative	ADJ
ap-1406	282	17	beta	beta	NOUN
ap-1406	282	18	transformation	transformation	NOUN
ap-1406	282	19	,	,	PUNCT
ap-1406	282	20	preprint	preprint	NOUN
ap-1406	282	21	2011	2011	NUM
ap-1406	282	22	,	,	PUNCT
ap-1406	282	23	18pp	18pp	ADJ
ap-1406	282	24	.	.	PUNCT
ap-1406	283	1	http://arxiv.org/abs/1101.2366	http://arxiv.org/abs/1101.2366	PROPN
ap-1406	284	1	[	[	X
ap-1406	284	2	10	10	NUM
ap-1406	284	3	]	]	X
ap-1406	284	4	masáková	masáková	PROPN
ap-1406	284	5	,	,	PUNCT
ap-1406	284	6	z.	z.	PROPN
ap-1406	284	7	,	,	PUNCT
ap-1406	284	8	pelantová	pelantová	PROPN
ap-1406	284	9	,	,	PUNCT
ap-1406	284	10	e.	e.	PROPN
ap-1406	284	11	,	,	PUNCT
ap-1406	284	12	vávra	vávra	PROPN
ap-1406	284	13	,	,	PUNCT
ap-1406	284	14	t.	t.	PROPN
ap-1406	284	15	:	:	PUNCT
ap-1406	284	16	arithmetics	arithmetic	NOUN
ap-1406	284	17	in	in	ADP
ap-1406	284	18	number	number	NOUN
ap-1406	284	19	systems	system	NOUN
ap-1406	284	20	with	with	ADP
ap-1406	284	21	negative	negative	ADJ
ap-1406	284	22	base	base	NOUN
ap-1406	284	23	,	,	PUNCT
ap-1406	284	24	theor	theor	PROPN
ap-1406	284	25	.	.	PUNCT
ap-1406	285	1	comp	comp	PROPN
ap-1406	285	2	.	.	PUNCT
ap-1406	286	1	sci	sci	PROPN
ap-1406	286	2	.	.	PROPN
ap-1406	287	1	412	412	NUM
ap-1406	287	2	(	(	PUNCT
ap-1406	287	3	2011	2011	NUM
ap-1406	287	4	)	)	PUNCT
ap-1406	287	5	,	,	PUNCT
ap-1406	287	6	835–845	835–845	NUM
ap-1406	287	7	.	.	PUNCT
ap-1406	288	1	[	[	X
ap-1406	288	2	11	11	NUM
ap-1406	288	3	]	]	X
ap-1406	288	4	parry	parry	PROPN
ap-1406	288	5	,	,	PUNCT
ap-1406	288	6	w.	w.	PROPN
ap-1406	288	7	:	:	PUNCT
ap-1406	288	8	on	on	ADP
ap-1406	288	9	the	the	DET
ap-1406	288	10	β	β	NOUN
ap-1406	288	11	-	-	NOUN
ap-1406	288	12	expansions	expansion	NOUN
ap-1406	288	13	of	of	ADP
ap-1406	288	14	real	real	ADJ
ap-1406	288	15	numbers	number	NOUN
ap-1406	288	16	,	,	PUNCT
ap-1406	288	17	acta	acta	PROPN
ap-1406	288	18	math	math	PROPN
ap-1406	288	19	.	.	PUNCT
ap-1406	289	1	acad	acad	PROPN
ap-1406	289	2	.	.	PUNCT
ap-1406	290	1	sci	sci	PROPN
ap-1406	290	2	.	.	PROPN
ap-1406	290	3	hung	hung	PROPN
ap-1406	290	4	.	.	PROPN
ap-1406	291	1	11	11	NUM
ap-1406	291	2	(	(	PUNCT
ap-1406	291	3	1960	1960	NUM
ap-1406	291	4	)	)	PUNCT
ap-1406	291	5	,	,	PUNCT
ap-1406	292	1	401–416	401–416	NUM
ap-1406	292	2	.	.	PUNCT
ap-1406	293	1	[	[	X
ap-1406	293	2	12	12	NUM
ap-1406	293	3	]	]	X
ap-1406	293	4	rényi	rényi	PROPN
ap-1406	293	5	,	,	PUNCT
ap-1406	293	6	a.	a.	NOUN
ap-1406	293	7	:	:	PUNCT
ap-1406	293	8	representations	representation	NOUN
ap-1406	293	9	for	for	ADP
ap-1406	293	10	real	real	ADJ
ap-1406	293	11	numbers	number	NOUN
ap-1406	293	12	and	and	CCONJ
ap-1406	293	13	their	their	PRON
ap-1406	293	14	ergodic	ergodic	ADJ
ap-1406	293	15	properties	property	NOUN
ap-1406	293	16	,	,	PUNCT
ap-1406	293	17	acta	acta	PROPN
ap-1406	293	18	math	math	PROPN
ap-1406	293	19	.	.	PUNCT
ap-1406	294	1	acad	acad	PROPN
ap-1406	294	2	.	.	PUNCT
ap-1406	295	1	sci	sci	PROPN
ap-1406	295	2	.	.	PROPN
ap-1406	295	3	hung	hung	PROPN
ap-1406	295	4	.	.	PROPN
ap-1406	296	1	8	8	NUM
ap-1406	296	2	(	(	PUNCT
ap-1406	296	3	1957	1957	NUM
ap-1406	296	4	)	)	PUNCT
ap-1406	296	5	,	,	PUNCT
ap-1406	297	1	477–493	477–493	NUM
ap-1406	297	2	.	.	PUNCT
ap-1406	298	1	[	[	X
ap-1406	298	2	13	13	NUM
ap-1406	298	3	]	]	PUNCT
ap-1406	298	4	schmidt	schmidt	NOUN
ap-1406	298	5	,	,	PUNCT
ap-1406	298	6	k.	k.	PROPN
ap-1406	298	7	:	:	PUNCT
ap-1406	298	8	on	on	ADP
ap-1406	298	9	periodic	periodic	ADJ
ap-1406	298	10	expansions	expansion	NOUN
ap-1406	298	11	of	of	ADP
ap-1406	298	12	pisot	pisot	ADJ
ap-1406	298	13	numbers	number	NOUN
ap-1406	298	14	and	and	CCONJ
ap-1406	298	15	salem	salem	NOUN
ap-1406	298	16	numbers	number	NOUN
ap-1406	298	17	,	,	PUNCT
ap-1406	298	18	bull	bull	NOUN
ap-1406	298	19	.	.	PUNCT
ap-1406	299	1	london	london	PROPN
ap-1406	299	2	math	math	PROPN
ap-1406	299	3	.	.	PUNCT
ap-1406	300	1	soc	soc	PROPN
ap-1406	300	2	.	.	PUNCT
ap-1406	301	1	12	12	NUM
ap-1406	301	2	(	(	PUNCT
ap-1406	301	3	1980	1980	NUM
ap-1406	301	4	)	)	PUNCT
ap-1406	301	5	,	,	PUNCT
ap-1406	301	6	269–278	269–278	NUM
ap-1406	301	7	.	.	PUNCT
ap-1406	302	1	[	[	X
ap-1406	302	2	14	14	NUM
ap-1406	302	3	]	]	X
ap-1406	302	4	solomyak	solomyak	PROPN
ap-1406	302	5	,	,	PUNCT
ap-1406	302	6	b.	b.	NOUN
ap-1406	302	7	:	:	PUNCT
ap-1406	302	8	conjugates	conjugate	NOUN
ap-1406	302	9	of	of	ADP
ap-1406	302	10	beta	beta	NOUN
ap-1406	302	11	-	-	PUNCT
ap-1406	302	12	numbers	number	NOUN
ap-1406	302	13	and	and	CCONJ
ap-1406	302	14	the	the	DET
ap-1406	302	15	zero	zero	NUM
ap-1406	302	16	-	-	PUNCT
ap-1406	302	17	free	free	ADJ
ap-1406	302	18	domain	domain	NOUN
ap-1406	302	19	for	for	ADP
ap-1406	302	20	a	a	DET
ap-1406	302	21	class	class	NOUN
ap-1406	302	22	of	of	ADP
ap-1406	302	23	analytic	analytic	ADJ
ap-1406	302	24	functions	function	NOUN
ap-1406	302	25	,	,	PUNCT
ap-1406	302	26	proc	proc	NOUN
ap-1406	302	27	.	.	PUNCT
ap-1406	303	1	london	london	PROPN
ap-1406	303	2	math	math	PROPN
ap-1406	303	3	.	.	PUNCT
ap-1406	304	1	soc	soc	PROPN
ap-1406	304	2	.	.	PUNCT
ap-1406	305	1	68	68	NUM
ap-1406	305	2	(	(	PUNCT
ap-1406	305	3	1994	1994	NUM
ap-1406	305	4	)	)	PUNCT
ap-1406	305	5	,	,	PUNCT
ap-1406	305	6	477–498	477–498	NUM
ap-1406	305	7	.	.	PUNCT
ap-1406	306	1	[	[	X
ap-1406	306	2	15	15	NUM
ap-1406	306	3	]	]	X
ap-1406	306	4	thurston	thurston	PROPN
ap-1406	306	5	,	,	PUNCT
ap-1406	306	6	w.	w.	PROPN
ap-1406	306	7	p.	p.	PROPN
ap-1406	306	8	:	:	PUNCT
ap-1406	306	9	groups	group	NOUN
ap-1406	306	10	,	,	PUNCT
ap-1406	306	11	tilings	tiling	NOUN
ap-1406	306	12	,	,	PUNCT
ap-1406	306	13	and	and	CCONJ
ap-1406	306	14	finite	finite	VERB
ap-1406	306	15	state	state	NOUN
ap-1406	306	16	automata	automata	NOUN
ap-1406	306	17	,	,	PUNCT
ap-1406	306	18	ams	am	NOUN
ap-1406	306	19	colloquium	colloquium	NOUN
ap-1406	306	20	lecture	lecture	NOUN
ap-1406	306	21	notes	note	NOUN
ap-1406	306	22	,	,	PUNCT
ap-1406	306	23	american	american	PROPN
ap-1406	306	24	mathematical	mathematical	ADJ
ap-1406	306	25	society	society	NOUN
ap-1406	306	26	,	,	PUNCT
ap-1406	306	27	boulder	boulder	NOUN
ap-1406	306	28	,	,	PUNCT
ap-1406	306	29	1989	1989	NUM
ap-1406	306	30	.	.	PUNCT
ap-1406	307	1	zuzana	zuzana	PROPN
ap-1406	307	2	masáková	masáková	PROPN
ap-1406	308	1	e	e	PROPN
ap-1406	308	2	-	-	NOUN
ap-1406	308	3	mail	mail	NOUN
ap-1406	308	4	:	:	PUNCT
ap-1406	308	5	zuzana.masakova@fjfi.cvut.cz	zuzana.masakova@fjfi.cvut.cz	PROPN
ap-1406	308	6	edita	edita	PROPN
ap-1406	308	7	pelantová	pelantová	PROPN
ap-1406	309	1	e	e	NOUN
ap-1406	309	2	-	-	NOUN
ap-1406	309	3	mail	mail	NOUN
ap-1406	309	4	:	:	PUNCT
ap-1406	309	5	edita.pelantova@fjfi.cvut.cz	edita.pelantova@fjfi.cvut.cz	PROPN
ap-1406	309	6	department	department	PROPN
ap-1406	309	7	of	of	ADP
ap-1406	309	8	mathematics	mathematics	PROPN
ap-1406	309	9	fnspe	fnspe	PROPN
ap-1406	309	10	czech	czech	PROPN
ap-1406	309	11	technical	technical	PROPN
ap-1406	309	12	university	university	PROPN
ap-1406	309	13	in	in	ADP
ap-1406	309	14	prague	prague	PROPN
ap-1406	309	15	trojanova	trojanova	X
ap-1406	309	16	13	13	NUM
ap-1406	309	17	,	,	PUNCT
ap-1406	309	18	120	120	NUM
ap-1406	309	19	00	00	NUM
ap-1406	309	20	praha	praha	PROPN
ap-1406	309	21	2	2	NUM
ap-1406	309	22	,	,	PUNCT
ap-1406	309	23	czech	czech	PROPN
ap-1406	309	24	republic	republic	NOUN
ap-1406	309	25	64	64	NUM