id	sid	tid	token	lemma	pos
ap-1255	1	1	ap-5-10.dvi	ap-5-10.dvi	PROPN
ap-1255	1	2	acta	acta	PROPN
ap-1255	1	3	polytechnica	polytechnica	PROPN
ap-1255	1	4	vol	vol	NOUN
ap-1255	1	5	.	.	PROPN
ap-1255	2	1	50	50	NUM
ap-1255	2	2	no	no	NOUN
ap-1255	2	3	.	.	PUNCT
ap-1255	3	1	5/2010	5/2010	DET
ap-1255	3	2	tilings	tiling	NOUN
ap-1255	3	3	generated	generate	VERB
ap-1255	3	4	by	by	ADP
ap-1255	3	5	ito	ito	PROPN
ap-1255	3	6	-	-	PROPN
ap-1255	3	7	sadahiro	sadahiro	PROPN
ap-1255	3	8	and	and	CCONJ
ap-1255	3	9	balanced	balanced	ADJ
ap-1255	3	10	(	(	PUNCT
ap-1255	3	11	−β)-numeration	−β)-numeration	NOUN
ap-1255	3	12	systems	system	NOUN
ap-1255	3	13	p.	p.	NOUN
ap-1255	3	14	ambrož	ambrož	PROPN
ap-1255	3	15	abstract	abstract	ADV
ap-1255	3	16	let	let	VERB
ap-1255	3	17	β	β	PRON
ap-1255	3	18	>	>	X
ap-1255	3	19	1	1	NUM
ap-1255	3	20	be	be	AUX
ap-1255	3	21	a	a	DET
ap-1255	3	22	cubic	cubic	ADJ
ap-1255	3	23	pisot	pisot	ADJ
ap-1255	3	24	unit	unit	NOUN
ap-1255	3	25	.	.	PUNCT
ap-1255	4	1	we	we	PRON
ap-1255	4	2	study	study	VERB
ap-1255	4	3	forms	form	NOUN
ap-1255	4	4	of	of	ADP
ap-1255	4	5	thurston	thurston	PROPN
ap-1255	4	6	tilings	tiling	NOUN
ap-1255	4	7	arising	arise	VERB
ap-1255	4	8	from	from	ADP
ap-1255	4	9	the	the	DET
ap-1255	4	10	classical	classical	ADJ
ap-1255	4	11	β	β	NOUN
ap-1255	4	12	-	-	ADJ
ap-1255	4	13	numeration	numeration	ADJ
ap-1255	4	14	system	system	NOUN
ap-1255	4	15	and	and	CCONJ
ap-1255	4	16	from	from	ADP
ap-1255	4	17	the	the	DET
ap-1255	4	18	(	(	PUNCT
ap-1255	4	19	−β)-numeration	−β)-numeration	NOUN
ap-1255	4	20	system	system	NOUN
ap-1255	4	21	for	for	ADP
ap-1255	4	22	both	both	CCONJ
ap-1255	4	23	the	the	DET
ap-1255	4	24	ito	ito	PROPN
ap-1255	4	25	-	-	PROPN
ap-1255	4	26	sadahiro	sadahiro	PROPN
ap-1255	4	27	and	and	CCONJ
ap-1255	4	28	balanced	balanced	ADJ
ap-1255	4	29	definition	definition	NOUN
ap-1255	4	30	of	of	ADP
ap-1255	4	31	the	the	DET
ap-1255	4	32	(	(	PUNCT
ap-1255	4	33	−β)-transformation	−β)-transformation	NOUN
ap-1255	4	34	.	.	PUNCT
ap-1255	5	1	keywords	keyword	NOUN
ap-1255	5	2	:	:	PUNCT
ap-1255	5	3	beta	beta	NOUN
ap-1255	5	4	-	-	PUNCT
ap-1255	5	5	expansion	expansion	NOUN
ap-1255	5	6	,	,	PUNCT
ap-1255	5	7	negative	negative	ADJ
ap-1255	5	8	base	base	NOUN
ap-1255	5	9	,	,	PUNCT
ap-1255	5	10	tiling	tile	VERB
ap-1255	5	11	.	.	PUNCT
ap-1255	6	1	1	1	NUM
ap-1255	6	2	introduction	introduction	NOUN
ap-1255	6	3	representations	representation	NOUN
ap-1255	6	4	of	of	ADP
ap-1255	6	5	real	real	ADJ
ap-1255	6	6	numbers	number	NOUN
ap-1255	6	7	in	in	ADP
ap-1255	6	8	a	a	DET
ap-1255	6	9	positional	positional	ADJ
ap-1255	6	10	numeration	numeration	NOUN
ap-1255	6	11	system	system	NOUN
ap-1255	6	12	with	with	ADP
ap-1255	6	13	an	an	DET
ap-1255	6	14	arbitrary	arbitrary	ADJ
ap-1255	6	15	base	base	NOUN
ap-1255	6	16	β	β	X
ap-1255	6	17	>	>	X
ap-1255	6	18	1	1	NUM
ap-1255	6	19	,	,	PUNCT
ap-1255	6	20	so	so	ADV
ap-1255	6	21	-	-	PUNCT
ap-1255	6	22	called	call	VERB
ap-1255	6	23	β	β	NOUN
ap-1255	6	24	-	-	NOUN
ap-1255	6	25	expansions	expansion	NOUN
ap-1255	6	26	,	,	PUNCT
ap-1255	6	27	were	be	AUX
ap-1255	6	28	introduced	introduce	VERB
ap-1255	6	29	by	by	ADP
ap-1255	6	30	rényi	rényi	PROPN
ap-1255	7	1	[	[	X
ap-1255	7	2	10	10	NUM
ap-1255	7	3	]	]	PUNCT
ap-1255	7	4	.	.	PUNCT
ap-1255	8	1	during	during	ADP
ap-1255	8	2	the	the	DET
ap-1255	8	3	fifty	fifty	NUM
ap-1255	8	4	years	year	NOUN
ap-1255	8	5	since	since	SCONJ
ap-1255	8	6	the	the	DET
ap-1255	8	7	publication	publication	NOUN
ap-1255	8	8	of	of	ADP
ap-1255	8	9	this	this	DET
ap-1255	8	10	seminal	seminal	ADJ
ap-1255	8	11	paper	paper	NOUN
ap-1255	8	12	,	,	PUNCT
ap-1255	8	13	β	β	NOUN
ap-1255	8	14	-	-	NOUN
ap-1255	8	15	expansions	expansion	NOUN
ap-1255	8	16	have	have	AUX
ap-1255	8	17	been	be	AUX
ap-1255	8	18	extensively	extensively	ADV
ap-1255	8	19	studied	study	VERB
ap-1255	8	20	from	from	ADP
ap-1255	8	21	various	various	ADJ
ap-1255	8	22	points	point	NOUN
ap-1255	8	23	of	of	ADP
ap-1255	8	24	view	view	NOUN
ap-1255	8	25	.	.	PUNCT
ap-1255	9	1	this	this	DET
ap-1255	9	2	paper	paper	NOUN
ap-1255	9	3	considers	consider	VERB
ap-1255	9	4	tilings	tiling	NOUN
ap-1255	9	5	generated	generate	VERB
ap-1255	9	6	by	by	ADP
ap-1255	9	7	β	β	NOUN
ap-1255	9	8	-	-	NOUN
ap-1255	9	9	expansions	expansion	NOUN
ap-1255	9	10	in	in	ADP
ap-1255	9	11	the	the	DET
ap-1255	9	12	case	case	NOUN
ap-1255	9	13	when	when	SCONJ
ap-1255	9	14	β	β	X
ap-1255	9	15	is	be	AUX
ap-1255	9	16	a	a	DET
ap-1255	9	17	pisot	pisot	ADJ
ap-1255	9	18	unit	unit	NOUN
ap-1255	9	19	.	.	PUNCT
ap-1255	10	1	a	a	DET
ap-1255	10	2	general	general	ADJ
ap-1255	10	3	method	method	NOUN
ap-1255	10	4	for	for	ADP
ap-1255	10	5	constructing	construct	VERB
ap-1255	10	6	the	the	DET
ap-1255	10	7	tiling	tiling	NOUN
ap-1255	10	8	of	of	ADP
ap-1255	10	9	a	a	DET
ap-1255	10	10	euclidean	euclidean	ADJ
ap-1255	10	11	space	space	NOUN
ap-1255	10	12	by	by	ADP
ap-1255	10	13	a	a	DET
ap-1255	10	14	pisot	pisot	ADJ
ap-1255	10	15	unit	unit	NOUN
ap-1255	10	16	was	be	AUX
ap-1255	10	17	proposed	propose	VERB
ap-1255	10	18	by	by	ADP
ap-1255	10	19	thurston	thurston	PROPN
ap-1255	11	1	[	[	X
ap-1255	11	2	11	11	NUM
ap-1255	11	3	]	]	PUNCT
ap-1255	11	4	,	,	PUNCT
ap-1255	11	5	although	although	SCONJ
ap-1255	11	6	an	an	DET
ap-1255	11	7	example	example	NOUN
ap-1255	11	8	of	of	ADP
ap-1255	11	9	such	such	DET
ap-1255	11	10	a	a	DET
ap-1255	11	11	tiling	tiling	NOUN
ap-1255	11	12	had	have	AUX
ap-1255	11	13	already	already	ADV
ap-1255	11	14	appeared	appear	VERB
ap-1255	11	15	in	in	ADP
ap-1255	11	16	the	the	DET
ap-1255	11	17	work	work	NOUN
ap-1255	11	18	of	of	ADP
ap-1255	11	19	rauzy	rauzy	NOUN
ap-1255	11	20	[	[	X
ap-1255	11	21	9	9	NUM
ap-1255	11	22	]	]	PUNCT
ap-1255	11	23	.	.	PUNCT
ap-1255	12	1	fundamental	fundamental	ADJ
ap-1255	12	2	properties	property	NOUN
ap-1255	12	3	of	of	ADP
ap-1255	12	4	these	these	DET
ap-1255	12	5	tilings	tiling	NOUN
ap-1255	12	6	were	be	AUX
ap-1255	12	7	later	later	ADV
ap-1255	12	8	studied	study	VERB
ap-1255	12	9	by	by	ADP
ap-1255	12	10	praggastis	praggastis	PROPN
ap-1255	12	11	[	[	X
ap-1255	12	12	8	8	NUM
ap-1255	12	13	]	]	PUNCT
ap-1255	12	14	and	and	CCONJ
ap-1255	12	15	akiyama	akiyama	PROPN
ap-1255	13	1	[	[	X
ap-1255	13	2	1	1	NUM
ap-1255	13	3	,	,	PUNCT
ap-1255	13	4	2	2	NUM
ap-1255	13	5	]	]	PUNCT
ap-1255	13	6	.	.	PUNCT
ap-1255	14	1	in	in	ADP
ap-1255	14	2	2009	2009	NUM
ap-1255	14	3	,	,	PUNCT
ap-1255	14	4	ito	ito	PROPN
ap-1255	14	5	and	and	CCONJ
ap-1255	14	6	sadahiro	sadahiro	PROPN
ap-1255	14	7	introduced	introduce	VERB
ap-1255	14	8	a	a	DET
ap-1255	14	9	new	new	ADJ
ap-1255	14	10	numeration	numeration	NOUN
ap-1255	14	11	system	system	NOUN
ap-1255	14	12	[	[	X
ap-1255	14	13	6	6	NUM
ap-1255	14	14	]	]	PUNCT
ap-1255	14	15	,	,	PUNCT
ap-1255	14	16	using	use	VERB
ap-1255	14	17	a	a	DET
ap-1255	14	18	non	non	ADJ
ap-1255	14	19	-	-	ADJ
ap-1255	14	20	integer	integer	ADJ
ap-1255	14	21	negative	negative	ADJ
ap-1255	14	22	base	base	NOUN
ap-1255	14	23	−β	−β	NOUN
ap-1255	14	24	<	<	X
ap-1255	14	25	−1	−1	NOUN
ap-1255	14	26	.	.	PUNCT
ap-1255	15	1	their	their	PRON
ap-1255	15	2	approach	approach	NOUN
ap-1255	15	3	is	be	AUX
ap-1255	15	4	very	very	ADV
ap-1255	15	5	similar	similar	ADJ
ap-1255	15	6	to	to	ADP
ap-1255	15	7	the	the	DET
ap-1255	15	8	approach	approach	NOUN
ap-1255	15	9	by	by	ADP
ap-1255	15	10	rényi	rényi	PROPN
ap-1255	15	11	.	.	PUNCT
ap-1255	16	1	another	another	DET
ap-1255	16	2	definition	definition	NOUN
ap-1255	16	3	of	of	ADP
ap-1255	16	4	a	a	DET
ap-1255	16	5	system	system	NOUN
ap-1255	16	6	using	use	VERB
ap-1255	16	7	a	a	DET
ap-1255	16	8	non	non	ADJ
ap-1255	16	9	-	-	ADJ
ap-1255	16	10	integer	integer	ADJ
ap-1255	16	11	negative	negative	ADJ
ap-1255	16	12	base	base	NOUN
ap-1255	16	13	−β	−β	PROPN
ap-1255	16	14	<	<	X
ap-1255	16	15	−1	−1	NOUN
ap-1255	16	16	,	,	PUNCT
ap-1255	16	17	obtained	obtain	VERB
ap-1255	16	18	as	as	ADP
ap-1255	16	19	a	a	DET
ap-1255	16	20	slight	slight	ADJ
ap-1255	16	21	modification	modification	NOUN
ap-1255	16	22	of	of	ADP
ap-1255	16	23	the	the	DET
ap-1255	16	24	system	system	NOUN
ap-1255	16	25	by	by	ADP
ap-1255	16	26	ito	ito	PROPN
ap-1255	16	27	and	and	CCONJ
ap-1255	16	28	sadahiro	sadahiro	PROPN
ap-1255	16	29	,	,	PUNCT
ap-1255	16	30	was	be	AUX
ap-1255	16	31	considered	consider	VERB
ap-1255	16	32	by	by	ADP
ap-1255	16	33	dombek	dombek	PROPN
ap-1255	16	34	[	[	X
ap-1255	16	35	4	4	NUM
ap-1255	16	36	]	]	PUNCT
ap-1255	16	37	.	.	PUNCT
ap-1255	17	1	the	the	DET
ap-1255	17	2	main	main	ADJ
ap-1255	17	3	subject	subject	NOUN
ap-1255	17	4	of	of	ADP
ap-1255	17	5	this	this	DET
ap-1255	17	6	paper	paper	NOUN
ap-1255	17	7	is	be	AUX
ap-1255	17	8	to	to	PART
ap-1255	17	9	transfer	transfer	VERB
ap-1255	17	10	the	the	DET
ap-1255	17	11	construction	construction	NOUN
ap-1255	17	12	by	by	ADP
ap-1255	17	13	thurston	thurston	PROPN
ap-1255	17	14	into	into	ADP
ap-1255	17	15	the	the	DET
ap-1255	17	16	framework	framework	NOUN
ap-1255	17	17	of	of	ADP
ap-1255	17	18	(	(	PUNCT
ap-1255	17	19	−β)numeration	−β)numeration	NOUN
ap-1255	17	20	(	(	PUNCT
ap-1255	17	21	both	both	DET
ap-1255	17	22	cases	case	NOUN
ap-1255	17	23	)	)	PUNCT
ap-1255	17	24	and	and	CCONJ
ap-1255	17	25	to	to	PART
ap-1255	17	26	provide	provide	VERB
ap-1255	17	27	examples	example	NOUN
ap-1255	17	28	of	of	ADP
ap-1255	17	29	how	how	SCONJ
ap-1255	17	30	tilings	tiling	NOUN
ap-1255	17	31	(	(	PUNCT
ap-1255	17	32	for	for	ADP
ap-1255	17	33	fixed	fix	VERB
ap-1255	17	34	β	β	NOUN
ap-1255	17	35	)	)	PUNCT
ap-1255	17	36	in	in	ADP
ap-1255	17	37	the	the	DET
ap-1255	17	38	positive	positive	ADJ
ap-1255	17	39	and	and	CCONJ
ap-1255	17	40	negative	negative	ADJ
ap-1255	17	41	case	case	NOUN
ap-1255	17	42	can	can	AUX
ap-1255	17	43	resemble	resemble	VERB
ap-1255	17	44	and/or	and/or	CCONJ
ap-1255	17	45	differ	differ	VERB
ap-1255	17	46	from	from	ADP
ap-1255	17	47	each	each	DET
ap-1255	17	48	other	other	ADJ
ap-1255	17	49	.	.	PUNCT
ap-1255	18	1	the	the	DET
ap-1255	18	2	paper	paper	NOUN
ap-1255	18	3	is	be	AUX
ap-1255	18	4	intended	intend	VERB
ap-1255	18	5	as	as	ADP
ap-1255	18	6	an	an	DET
ap-1255	18	7	entry	entry	NOUN
ap-1255	18	8	point	point	NOUN
ap-1255	18	9	into	into	ADP
ap-1255	18	10	a	a	DET
ap-1255	18	11	study	study	NOUN
ap-1255	18	12	of	of	ADP
ap-1255	18	13	the	the	DET
ap-1255	18	14	properties	property	NOUN
ap-1255	18	15	of	of	ADP
ap-1255	18	16	these	these	DET
ap-1255	18	17	tilings	tiling	NOUN
ap-1255	18	18	.	.	PUNCT
ap-1255	19	1	2	2	NUM
ap-1255	19	2	rényi	rényi	PROPN
ap-1255	19	3	β	β	NOUN
ap-1255	19	4	-	-	NOUN
ap-1255	19	5	expansions	expansion	NOUN
ap-1255	19	6	let	let	VERB
ap-1255	19	7	β	β	PRON
ap-1255	19	8	>	>	X
ap-1255	19	9	1	1	NUM
ap-1255	19	10	be	be	AUX
ap-1255	19	11	a	a	DET
ap-1255	19	12	real	real	ADJ
ap-1255	19	13	number	number	NOUN
ap-1255	19	14	and	and	CCONJ
ap-1255	19	15	let	let	VERB
ap-1255	19	16	the	the	DET
ap-1255	19	17	transformation	transformation	NOUN
ap-1255	19	18	tβ	tβ	NOUN
ap-1255	19	19	:	:	PUNCT
ap-1255	20	1	[	[	X
ap-1255	20	2	0	0	NUM
ap-1255	20	3	,	,	PUNCT
ap-1255	20	4	1	1	NUM
ap-1255	20	5	)	)	PUNCT
ap-1255	20	6	→	→	PUNCT
ap-1255	21	1	[	[	X
ap-1255	21	2	0	0	NUM
ap-1255	21	3	,	,	PUNCT
ap-1255	21	4	1	1	NUM
ap-1255	21	5	)	)	PUNCT
ap-1255	21	6	be	be	AUX
ap-1255	21	7	defined	define	VERB
ap-1255	21	8	by	by	ADP
ap-1255	21	9	the	the	DET
ap-1255	21	10	prescription	prescription	NOUN
ap-1255	21	11	tβ(x	tβ(x	PUNCT
ap-1255	21	12	)	)	PUNCT
ap-1255	21	13	:	:	PUNCT
ap-1255	21	14	=	=	PUNCT
ap-1255	21	15	βx	βx	ADP
ap-1255	21	16	−	−	PROPN
ap-1255	21	17	�	�	PROPN
ap-1255	21	18	βx	βx	NOUN
ap-1255	21	19	�	�	PROPN
ap-1255	21	20	.	.	PUNCT
ap-1255	22	1	the	the	DET
ap-1255	22	2	representation	representation	NOUN
ap-1255	22	3	of	of	ADP
ap-1255	22	4	a	a	DET
ap-1255	22	5	number	number	NOUN
ap-1255	22	6	x	x	SYM
ap-1255	22	7	∈	∈	PROPN
ap-1255	22	8	[	[	X
ap-1255	22	9	0	0	NUM
ap-1255	22	10	,	,	PUNCT
ap-1255	22	11	1	1	NUM
ap-1255	22	12	)	)	PUNCT
ap-1255	22	13	of	of	ADP
ap-1255	22	14	the	the	DET
ap-1255	22	15	form	form	NOUN
ap-1255	22	16	x	x	PUNCT
ap-1255	22	17	=	=	SYM
ap-1255	22	18	x1	x1	NUM
ap-1255	22	19	β	β	X
ap-1255	22	20	+	+	SYM
ap-1255	22	21	x2	x2	PROPN
ap-1255	22	22	β2	β2	NOUN
ap-1255	22	23	+	+	CCONJ
ap-1255	22	24	x3	x3	ADJ
ap-1255	22	25	β3	β3	ADJ
ap-1255	22	26	+	+	CCONJ
ap-1255	22	27	·	·	PUNCT
ap-1255	22	28	·	·	PUNCT
ap-1255	22	29	·	·	PUNCT
ap-1255	22	30	,	,	PUNCT
ap-1255	22	31	where	where	SCONJ
ap-1255	22	32	xi	xi	PROPN
ap-1255	22	33	=	=	PUNCT
ap-1255	22	34	�	�	PROPN
ap-1255	22	35	βt	βt	PROPN
ap-1255	22	36	i−1	i−1	PROPN
ap-1255	22	37	β	β	PROPN
ap-1255	22	38	(	(	PUNCT
ap-1255	22	39	x	x	NOUN
ap-1255	22	40	)	)	PUNCT
ap-1255	22	41	�	�	PROPN
ap-1255	22	42	,	,	PUNCT
ap-1255	22	43	is	be	AUX
ap-1255	22	44	called	call	VERB
ap-1255	22	45	the	the	DET
ap-1255	22	46	β	β	NOUN
ap-1255	22	47	-	-	NOUN
ap-1255	22	48	expansion	expansion	NOUN
ap-1255	22	49	of	of	ADP
ap-1255	22	50	x.	x.	NOUN
ap-1255	22	51	since	since	SCONJ
ap-1255	22	52	βt	βt	PROPN
ap-1255	22	53	(	(	PUNCT
ap-1255	22	54	x	x	X
ap-1255	22	55	)	)	PUNCT
ap-1255	22	56	∈	∈	PROPN
ap-1255	23	1	[	[	X
ap-1255	23	2	0	0	NUM
ap-1255	23	3	,	,	PUNCT
ap-1255	23	4	β	β	NOUN
ap-1255	23	5	)	)	PUNCT
ap-1255	23	6	the	the	DET
ap-1255	23	7	coefficients	coefficient	NOUN
ap-1255	23	8	xi	xi	X
ap-1255	23	9	(	(	PUNCT
ap-1255	23	10	called	call	VERB
ap-1255	23	11	digits	digit	NOUN
ap-1255	23	12	)	)	PUNCT
ap-1255	23	13	are	be	AUX
ap-1255	23	14	elements	element	NOUN
ap-1255	23	15	of	of	ADP
ap-1255	23	16	the	the	DET
ap-1255	23	17	set	set	NOUN
ap-1255	23	18	{	{	PUNCT
ap-1255	23	19	0	0	NUM
ap-1255	23	20	,	,	PUNCT
ap-1255	23	21	1	1	NUM
ap-1255	23	22	,	,	PUNCT
ap-1255	23	23	.	.	PUNCT
ap-1255	23	24	.	.	PUNCT
ap-1255	24	1	.	.	PUNCT
ap-1255	25	1	,	,	PUNCT
ap-1255	25	2	�	�	PROPN
ap-1255	25	3	β	β	NOUN
ap-1255	25	4	�	�	NOUN
ap-1255	25	5	−	−	NUM
ap-1255	25	6	1	1	NUM
ap-1255	25	7	}	}	PUNCT
ap-1255	25	8	.	.	PUNCT
ap-1255	26	1	the	the	DET
ap-1255	26	2	β	β	NOUN
ap-1255	26	3	-	-	NOUN
ap-1255	26	4	expansion	expansion	NOUN
ap-1255	26	5	of	of	ADP
ap-1255	26	6	an	an	DET
ap-1255	26	7	arbitrary	arbitrary	ADJ
ap-1255	26	8	real	real	ADJ
ap-1255	26	9	number	number	NOUN
ap-1255	26	10	x	x	SYM
ap-1255	26	11	≥	≥	NOUN
ap-1255	26	12	1	1	NUM
ap-1255	26	13	can	can	AUX
ap-1255	26	14	be	be	AUX
ap-1255	26	15	naturally	naturally	ADV
ap-1255	26	16	defined	define	VERB
ap-1255	26	17	in	in	ADP
ap-1255	26	18	the	the	DET
ap-1255	26	19	following	following	ADJ
ap-1255	26	20	way	way	NOUN
ap-1255	26	21	:	:	PUNCT
ap-1255	26	22	find	find	VERB
ap-1255	26	23	an	an	DET
ap-1255	26	24	exponent	exponent	NOUN
ap-1255	26	25	k	k	PROPN
ap-1255	26	26	∈	∈	PROPN
ap-1255	26	27	n	n	PRON
ap-1255	26	28	such	such	ADJ
ap-1255	26	29	that	that	SCONJ
ap-1255	26	30	x	x	PUNCT
ap-1255	26	31	βk	βk	ADP
ap-1255	26	32	∈	∈	PROPN
ap-1255	27	1	[	[	X
ap-1255	27	2	0	0	NUM
ap-1255	27	3	,	,	PUNCT
ap-1255	27	4	1	1	NUM
ap-1255	27	5	)	)	PUNCT
ap-1255	27	6	.	.	PUNCT
ap-1255	28	1	using	use	VERB
ap-1255	28	2	the	the	DET
ap-1255	28	3	transformation	transformation	NOUN
ap-1255	28	4	tβ	tβ	PRON
ap-1255	28	5	derive	derive	VERB
ap-1255	28	6	the	the	DET
ap-1255	28	7	β	β	NOUN
ap-1255	28	8	-	-	NOUN
ap-1255	28	9	expansion	expansion	NOUN
ap-1255	28	10	of	of	ADP
ap-1255	28	11	x	x	PUNCT
ap-1255	28	12	βk	βk	NOUN
ap-1255	28	13	of	of	ADP
ap-1255	28	14	the	the	DET
ap-1255	28	15	form	form	NOUN
ap-1255	28	16	x	x	PUNCT
ap-1255	28	17	βk	βk	ADP
ap-1255	28	18	=	=	SYM
ap-1255	28	19	x1	x1	ADJ
ap-1255	28	20	β	β	X
ap-1255	28	21	+	+	CCONJ
ap-1255	28	22	x2	x2	PROPN
ap-1255	28	23	β2	β2	NOUN
ap-1255	28	24	+	+	CCONJ
ap-1255	28	25	x3	x3	ADJ
ap-1255	28	26	β3	β3	ADJ
ap-1255	28	27	+	+	PRON
ap-1255	28	28	.	.	PUNCT
ap-1255	28	29	.	.	PUNCT
ap-1255	28	30	.	.	PUNCT
ap-1255	29	1	,	,	PUNCT
ap-1255	29	2	so	so	SCONJ
ap-1255	29	3	that	that	SCONJ
ap-1255	29	4	x	x	NOUN
ap-1255	29	5	=	=	PUNCT
ap-1255	29	6	x1β	x1β	PROPN
ap-1255	29	7	k−1	k−1	PROPN
ap-1255	29	8	+	+	PUNCT
ap-1255	29	9	x2β	x2β	PROPN
ap-1255	29	10	k−2	k−2	PROPN
ap-1255	29	11	+	+	X
ap-1255	29	12	.	.	PUNCT
ap-1255	29	13	.	.	PUNCT
ap-1255	30	1	.+	.+	NOUN
ap-1255	30	2	xk−1β	xk−1β	PROPN
ap-1255	31	1	+	+	CCONJ
ap-1255	31	2	xk	xk	PROPN
ap-1255	32	1	+	+	CCONJ
ap-1255	32	2	xk+1	xk+1	PROPN
ap-1255	32	3	β	β	X
ap-1255	32	4	+	+	X
ap-1255	32	5	.	.	PUNCT
ap-1255	32	6	.	.	PUNCT
ap-1255	32	7	.	.	PUNCT
ap-1255	33	1	the	the	DET
ap-1255	33	2	β	β	NOUN
ap-1255	33	3	-	-	NOUN
ap-1255	33	4	expansion	expansion	NOUN
ap-1255	33	5	of	of	ADP
ap-1255	33	6	x	x	SYM
ap-1255	33	7	∈	∈	PROPN
ap-1255	33	8	r+	r+	NOUN
ap-1255	33	9	is	be	AUX
ap-1255	33	10	denoted	denote	VERB
ap-1255	33	11	by	by	ADP
ap-1255	33	12	dβ(x	dβ(x	NOUN
ap-1255	33	13	)	)	PUNCT
ap-1255	33	14	,	,	PUNCT
ap-1255	33	15	and	and	CCONJ
ap-1255	33	16	as	as	ADP
ap-1255	33	17	usual	usual	ADJ
ap-1255	33	18	we	we	PRON
ap-1255	33	19	write	write	VERB
ap-1255	33	20	dβ(x	dβ(x	NOUN
ap-1255	33	21	)	)	PUNCT
ap-1255	33	22	=	=	SYM
ap-1255	34	1	x1x2	x1x2	X
ap-1255	34	2	.	.	PUNCT
ap-1255	34	3	.	.	PUNCT
ap-1255	34	4	.	.	PUNCT
ap-1255	35	1	xk•xk+1xk+2	xk•xk+1xk+2	PUNCT
ap-1255	35	2	.	.	PUNCT
ap-1255	35	3	.	.	PUNCT
ap-1255	36	1	.	.	PUNCT
ap-1255	37	1	the	the	DET
ap-1255	37	2	digit	digit	NOUN
ap-1255	37	3	string	string	PROPN
ap-1255	37	4	x1x2x3	x1x2x3	PROPN
ap-1255	37	5	·	·	PUNCT
ap-1255	37	6	·	·	PUNCT
ap-1255	37	7	·	·	PUNCT
ap-1255	37	8	is	be	AUX
ap-1255	37	9	said	say	VERB
ap-1255	37	10	to	to	PART
ap-1255	37	11	be	be	AUX
ap-1255	37	12	β	β	NOUN
ap-1255	37	13	-	-	ADJ
ap-1255	37	14	admissible	admissible	ADJ
ap-1255	37	15	if	if	SCONJ
ap-1255	37	16	there	there	PRON
ap-1255	37	17	exists	exist	VERB
ap-1255	37	18	a	a	DET
ap-1255	37	19	number	number	NOUN
ap-1255	37	20	x	x	SYM
ap-1255	37	21	∈	∈	PROPN
ap-1255	38	1	[	[	X
ap-1255	38	2	0	0	NUM
ap-1255	38	3	,	,	PUNCT
ap-1255	38	4	1	1	NUM
ap-1255	38	5	)	)	PUNCT
ap-1255	38	6	so	so	SCONJ
ap-1255	38	7	that	that	SCONJ
ap-1255	38	8	dβ(x	dβ(x	VERB
ap-1255	38	9	)	)	PUNCT
ap-1255	38	10	=	=	SYM
ap-1255	39	1	•x1x2x3	•x1x2x3	PROPN
ap-1255	39	2	.	.	PUNCT
ap-1255	39	3	.	.	PUNCT
ap-1255	39	4	.	.	PUNCT
ap-1255	40	1	is	be	AUX
ap-1255	40	2	its	its	PRON
ap-1255	40	3	β	β	NOUN
ap-1255	40	4	-	-	NOUN
ap-1255	40	5	expansion	expansion	NOUN
ap-1255	40	6	.	.	PUNCT
ap-1255	41	1	the	the	DET
ap-1255	41	2	set	set	NOUN
ap-1255	41	3	of	of	ADP
ap-1255	41	4	admissible	admissible	ADJ
ap-1255	41	5	digit	digit	NOUN
ap-1255	41	6	strings	string	NOUN
ap-1255	41	7	can	can	AUX
ap-1255	41	8	be	be	AUX
ap-1255	41	9	described	describe	VERB
ap-1255	41	10	using	use	VERB
ap-1255	41	11	the	the	DET
ap-1255	41	12	rényi	rényi	PROPN
ap-1255	41	13	expansion	expansion	NOUN
ap-1255	41	14	7	7	NUM
ap-1255	41	15	acta	acta	PROPN
ap-1255	41	16	polytechnica	polytechnica	PROPN
ap-1255	41	17	vol	vol	NOUN
ap-1255	41	18	.	.	PUNCT
ap-1255	42	1	50	50	NUM
ap-1255	42	2	no	no	NOUN
ap-1255	42	3	.	.	PUNCT
ap-1255	43	1	5/2010	5/2010	NUM
ap-1255	43	2	of	of	ADP
ap-1255	43	3	1	1	NUM
ap-1255	43	4	,	,	PUNCT
ap-1255	43	5	denoted	denote	VERB
ap-1255	43	6	by	by	ADP
ap-1255	43	7	dβ(1	dβ(1	PROPN
ap-1255	43	8	)	)	PUNCT
ap-1255	43	9	=	=	NOUN
ap-1255	43	10	t1t2t3	t1t2t3	NOUN
ap-1255	43	11	.	.	PUNCT
ap-1255	43	12	.	.	PUNCT
ap-1255	44	1	.	.	PUNCT
ap-1255	45	1	,	,	PUNCT
ap-1255	45	2	where	where	SCONJ
ap-1255	45	3	t1	t1	NOUN
ap-1255	45	4	=	=	SYM
ap-1255	45	5	�	�	PROPN
ap-1255	45	6	β	β	NOUN
ap-1255	45	7	�	�	PROPN
ap-1255	45	8	and	and	CCONJ
ap-1255	45	9	dβ(β	dβ(β	NOUN
ap-1255	45	10	−	−	PROPN
ap-1255	45	11	�	�	PROPN
ap-1255	45	12	β	β	NOUN
ap-1255	45	13	�	�	NOUN
ap-1255	45	14	)	)	PUNCT
ap-1255	45	15	=	=	VERB
ap-1255	45	16	t2t3t4	t2t3t4	PROPN
ap-1255	45	17	.	.	PUNCT
ap-1255	45	18	.	.	PUNCT
ap-1255	46	1	..	..	PUNCT
ap-1255	47	1	the	the	DET
ap-1255	47	2	rényi	rényi	PROPN
ap-1255	47	3	expansion	expansion	NOUN
ap-1255	47	4	of	of	ADP
ap-1255	47	5	1	1	NUM
ap-1255	47	6	may	may	AUX
ap-1255	47	7	or	or	CCONJ
ap-1255	47	8	may	may	AUX
ap-1255	47	9	not	not	PART
ap-1255	47	10	be	be	AUX
ap-1255	47	11	finite	finite	ADJ
ap-1255	47	12	(	(	PUNCT
ap-1255	47	13	i.e.	i.e.	X
ap-1255	47	14	,	,	PUNCT
ap-1255	47	15	ending	end	VERB
ap-1255	47	16	in	in	ADP
ap-1255	47	17	infinitely	infinitely	ADV
ap-1255	47	18	many	many	ADJ
ap-1255	47	19	0	0	NUM
ap-1255	47	20	’s	’s	NOUN
ap-1255	47	21	which	which	PRON
ap-1255	47	22	are	be	AUX
ap-1255	47	23	omitted	omit	VERB
ap-1255	47	24	)	)	PUNCT
ap-1255	47	25	.	.	PUNCT
ap-1255	48	1	the	the	DET
ap-1255	48	2	infinite	infinite	ADJ
ap-1255	48	3	rényi	rényi	PROPN
ap-1255	48	4	expansion	expansion	NOUN
ap-1255	48	5	of	of	ADP
ap-1255	48	6	1	1	NUM
ap-1255	48	7	,	,	PUNCT
ap-1255	48	8	denoted	denote	VERB
ap-1255	48	9	by	by	ADP
ap-1255	48	10	d∗β(1	d∗β(1	NOUN
ap-1255	48	11	)	)	PUNCT
ap-1255	48	12	is	be	AUX
ap-1255	48	13	defined	define	VERB
ap-1255	48	14	by	by	ADP
ap-1255	48	15	d∗β(1	d∗β(1	NOUN
ap-1255	48	16	)	)	PUNCT
ap-1255	48	17	=	=	SYM
ap-1255	48	18	lim	lim	NOUN
ap-1255	48	19	ε→0	ε→0	NOUN
ap-1255	49	1	+	+	CCONJ
ap-1255	49	2	dβ(1−	dβ(1−	PROPN
ap-1255	49	3	ε	ε	PROPN
ap-1255	49	4	)	)	PUNCT
ap-1255	49	5	,	,	PUNCT
ap-1255	49	6	where	where	SCONJ
ap-1255	49	7	the	the	DET
ap-1255	49	8	limit	limit	NOUN
ap-1255	49	9	is	be	AUX
ap-1255	49	10	taken	take	VERB
ap-1255	49	11	over	over	ADP
ap-1255	49	12	the	the	DET
ap-1255	49	13	usual	usual	ADJ
ap-1255	49	14	product	product	NOUN
ap-1255	49	15	topology	topology	NOUN
ap-1255	49	16	on	on	ADP
ap-1255	49	17	{	{	PUNCT
ap-1255	49	18	0	0	NUM
ap-1255	49	19	,	,	PUNCT
ap-1255	49	20	1	1	NUM
ap-1255	49	21	,	,	PUNCT
ap-1255	49	22	.	.	PUNCT
ap-1255	49	23	.	.	PUNCT
ap-1255	49	24	.	.	PUNCT
ap-1255	50	1	,	,	PUNCT
ap-1255	50	2	�	�	PROPN
ap-1255	50	3	β	β	NOUN
ap-1255	50	4	�	�	PROPN
ap-1255	50	5	−	−	PROPN
ap-1255	50	6	1}n	1}n	NUM
ap-1255	50	7	.	.	PUNCT
ap-1255	51	1	it	it	PRON
ap-1255	51	2	can	can	AUX
ap-1255	51	3	be	be	AUX
ap-1255	51	4	shown	show	VERB
ap-1255	51	5	that	that	SCONJ
ap-1255	51	6	d∗β(1	d∗β(1	NOUN
ap-1255	51	7	)	)	PUNCT
ap-1255	51	8	=	=	NOUN
ap-1255	51	9	{	{	PUNCT
ap-1255	51	10	dβ(1	dβ(1	PROPN
ap-1255	51	11	)	)	PUNCT
ap-1255	51	12	if	if	SCONJ
ap-1255	51	13	dβ(1	dβ(1	PROPN
ap-1255	51	14	)	)	PUNCT
ap-1255	51	15	is	be	AUX
ap-1255	51	16	infinite	infinite	ADJ
ap-1255	51	17	,	,	PUNCT
ap-1255	51	18	(	(	PUNCT
ap-1255	51	19	t1	t1	NOUN
ap-1255	51	20	·	·	PUNCT
ap-1255	51	21	·	·	PUNCT
ap-1255	51	22	·	·	PUNCT
ap-1255	52	1	tm−1(tm	tm−1(tm	NOUN
ap-1255	52	2	−	−	NOUN
ap-1255	52	3	1	1	NUM
ap-1255	52	4	)	)	PUNCT
ap-1255	52	5	)	)	PUNCT
ap-1255	53	1	ω	ω	PROPN
ap-1255	53	2	if	if	SCONJ
ap-1255	53	3	dβ(1	dβ(1	PROPN
ap-1255	53	4	)	)	PUNCT
ap-1255	53	5	=	=	SYM
ap-1255	53	6	t1	t1	NOUN
ap-1255	53	7	·	·	PUNCT
ap-1255	53	8	·	·	PUNCT
ap-1255	53	9	·	·	PUNCT
ap-1255	53	10	tm0ω	tm0ω	PROPN
ap-1255	53	11	with	with	ADP
ap-1255	53	12	tm	tm	PROPN
ap-1255	53	13	=	=	NOUN
ap-1255	53	14	0	0	PROPN
ap-1255	53	15	.	.	PUNCT
ap-1255	54	1	the	the	DET
ap-1255	54	2	characterization	characterization	NOUN
ap-1255	54	3	of	of	ADP
ap-1255	54	4	admissible	admissible	ADJ
ap-1255	54	5	stings	sting	NOUN
ap-1255	54	6	is	be	AUX
ap-1255	54	7	given	give	VERB
ap-1255	54	8	by	by	ADP
ap-1255	54	9	the	the	DET
ap-1255	54	10	following	following	NOUN
ap-1255	54	11	theorem	theorem	NOUN
ap-1255	54	12	due	due	ADP
ap-1255	54	13	to	to	PART
ap-1255	54	14	parry	parry	VERB
ap-1255	54	15	.	.	PUNCT
ap-1255	55	1	theorem	theorem	ADJ
ap-1255	55	2	1	1	NUM
ap-1255	55	3	(	(	PUNCT
ap-1255	55	4	[	[	X
ap-1255	55	5	7	7	NUM
ap-1255	55	6	]	]	SYM
ap-1255	55	7	)	)	PUNCT
ap-1255	55	8	a	a	DET
ap-1255	55	9	string	string	NOUN
ap-1255	55	10	x1x2x3	x1x2x3	PROPN
ap-1255	55	11	.	.	PUNCT
ap-1255	55	12	.	.	PUNCT
ap-1255	55	13	.	.	PUNCT
ap-1255	56	1	over	over	ADP
ap-1255	56	2	the	the	DET
ap-1255	56	3	alphabet	alphabet	NOUN
ap-1255	56	4	{	{	PUNCT
ap-1255	56	5	0	0	NUM
ap-1255	56	6	,	,	PUNCT
ap-1255	56	7	1	1	NUM
ap-1255	56	8	,	,	PUNCT
ap-1255	56	9	.	.	PUNCT
ap-1255	56	10	.	.	PUNCT
ap-1255	56	11	.	.	PUNCT
ap-1255	57	1	,	,	PUNCT
ap-1255	57	2	�	�	PROPN
ap-1255	57	3	β	β	NOUN
ap-1255	57	4	�	�	NOUN
ap-1255	57	5	−	−	ADP
ap-1255	57	6	1	1	NUM
ap-1255	57	7	}	}	PUNCT
ap-1255	57	8	is	be	AUX
ap-1255	57	9	β	β	NOUN
ap-1255	57	10	-	-	ADJ
ap-1255	57	11	admissible	admissible	ADJ
ap-1255	57	12	,	,	PUNCT
ap-1255	57	13	if	if	SCONJ
ap-1255	57	14	and	and	CCONJ
ap-1255	57	15	only	only	ADV
ap-1255	57	16	if	if	SCONJ
ap-1255	57	17	for	for	ADP
ap-1255	57	18	all	all	DET
ap-1255	57	19	i	i	PRON
ap-1255	57	20	=	=	NOUN
ap-1255	57	21	1	1	NUM
ap-1255	57	22	,	,	PUNCT
ap-1255	57	23	2	2	NUM
ap-1255	57	24	,	,	PUNCT
ap-1255	57	25	3	3	NUM
ap-1255	57	26	,	,	PUNCT
ap-1255	57	27	.	.	PUNCT
ap-1255	57	28	.	.	PUNCT
ap-1255	58	1	.	.	PUNCT
ap-1255	59	1	,	,	PUNCT
ap-1255	59	2	0ω	0ω	PROPN
ap-1255	59	3	�	�	PROPN
ap-1255	59	4	lex	lex	PROPN
ap-1255	59	5	xixi+1xi+2	xixi+1xi+2	PROPN
ap-1255	59	6	.	.	PUNCT
ap-1255	59	7	.	.	PUNCT
ap-1255	59	8	.	.	PUNCT
ap-1255	60	1	≺lex	≺lex	NOUN
ap-1255	60	2	d∗β(1	d∗β(1	PROPN
ap-1255	60	3	)	)	PUNCT
ap-1255	60	4	,	,	PUNCT
ap-1255	60	5	where	where	SCONJ
ap-1255	60	6	�	�	PROPN
ap-1255	60	7	lex	lex	PROPN
ap-1255	60	8	is	be	AUX
ap-1255	60	9	the	the	DET
ap-1255	60	10	lexicographical	lexicographical	ADJ
ap-1255	60	11	order	order	NOUN
ap-1255	60	12	.	.	PUNCT
ap-1255	61	1	using	use	VERB
ap-1255	61	2	β	β	ADJ
ap-1255	61	3	-	-	ADJ
ap-1255	61	4	admissible	admissible	ADJ
ap-1255	61	5	digit	digit	NOUN
ap-1255	61	6	strings	string	NOUN
ap-1255	61	7	,	,	PUNCT
ap-1255	61	8	one	one	PRON
ap-1255	61	9	can	can	AUX
ap-1255	61	10	define	define	VERB
ap-1255	61	11	the	the	DET
ap-1255	61	12	set	set	NOUN
ap-1255	61	13	of	of	ADP
ap-1255	61	14	non	non	ADJ
ap-1255	61	15	-	-	ADJ
ap-1255	61	16	negative	negative	ADJ
ap-1255	61	17	β	β	NOUN
ap-1255	61	18	-	-	NOUN
ap-1255	61	19	integers	integer	NOUN
ap-1255	61	20	,	,	PUNCT
ap-1255	61	21	denoted	denote	VERB
ap-1255	61	22	zβ	zβ	PROPN
ap-1255	61	23	,	,	PUNCT
ap-1255	61	24	zβ	zβ	X
ap-1255	61	25	:	:	PUNCT
ap-1255	61	26	=	=	SYM
ap-1255	61	27	{	{	PUNCT
ap-1255	61	28	akβk	akβk	ADJ
ap-1255	61	29	+	+	X
ap-1255	61	30	.	.	PUNCT
ap-1255	61	31	.	.	PUNCT
ap-1255	61	32	.	.	PUNCT
ap-1255	62	1	a1β	a1β	PROPN
ap-1255	63	1	+	+	CCONJ
ap-1255	63	2	a0	a0	PROPN
ap-1255	63	3	|	|	PROPN
ap-1255	63	4	ak	ak	PROPN
ap-1255	63	5	·	·	PUNCT
ap-1255	63	6	·	·	PUNCT
ap-1255	63	7	·	·	PUNCT
ap-1255	64	1	a1a00ω	a1a00ω	NOUN
ap-1255	64	2	is	be	AUX
ap-1255	64	3	a	a	DET
ap-1255	64	4	β	β	NOUN
ap-1255	64	5	-	-	ADJ
ap-1255	64	6	admissible	admissible	ADJ
ap-1255	64	7	digit	digit	NOUN
ap-1255	64	8	string	string	NOUN
ap-1255	64	9	}	}	PUNCT
ap-1255	64	10	,	,	PUNCT
ap-1255	64	11	and	and	CCONJ
ap-1255	64	12	the	the	DET
ap-1255	64	13	set	set	VERB
ap-1255	64	14	fin(β	fin(β	PROPN
ap-1255	64	15	)	)	PUNCT
ap-1255	64	16	of	of	ADP
ap-1255	64	17	those	those	DET
ap-1255	64	18	x	x	SYM
ap-1255	64	19	∈	∈	PROPN
ap-1255	64	20	r+	r+	PUNCT
ap-1255	64	21	whose	whose	DET
ap-1255	64	22	β	β	NOUN
ap-1255	64	23	-	-	NOUN
ap-1255	64	24	expansions	expansion	NOUN
ap-1255	64	25	have	have	VERB
ap-1255	64	26	only	only	ADV
ap-1255	64	27	finitely	finitely	ADV
ap-1255	64	28	many	many	ADJ
ap-1255	64	29	non	non	ADJ
ap-1255	64	30	-	-	ADJ
ap-1255	64	31	zero	zero	NUM
ap-1255	64	32	coefficients	coefficient	NOUN
ap-1255	64	33	to	to	ADP
ap-1255	64	34	the	the	DET
ap-1255	64	35	right	right	NOUN
ap-1255	64	36	from	from	ADP
ap-1255	64	37	the	the	DET
ap-1255	64	38	fractional	fractional	ADJ
ap-1255	64	39	point	point	NOUN
ap-1255	64	40	fin(β	fin(β	PROPN
ap-1255	64	41	)	)	PUNCT
ap-1255	64	42	:	:	PUNCT
ap-1255	65	1	=	=	SYM
ap-1255	65	2	⋃	⋃	NOUN
ap-1255	65	3	n∈n	n∈n	NOUN
ap-1255	65	4	1	1	NUM
ap-1255	65	5	βn	βn	PROPN
ap-1255	65	6	zβ	zβ	PROPN
ap-1255	65	7	.	.	PUNCT
ap-1255	66	1	the	the	DET
ap-1255	66	2	distances	distance	NOUN
ap-1255	66	3	between	between	ADP
ap-1255	66	4	consecutive	consecutive	ADJ
ap-1255	66	5	β	β	NOUN
ap-1255	66	6	-	-	NOUN
ap-1255	66	7	integers	integer	NOUN
ap-1255	66	8	are	be	AUX
ap-1255	66	9	described	describe	VERB
ap-1255	66	10	in	in	ADP
ap-1255	66	11	[	[	X
ap-1255	66	12	11	11	NUM
ap-1255	66	13	]	]	PUNCT
ap-1255	66	14	.	.	PUNCT
ap-1255	67	1	it	it	PRON
ap-1255	67	2	is	be	AUX
ap-1255	67	3	shown	show	VERB
ap-1255	67	4	that	that	SCONJ
ap-1255	67	5	they	they	PRON
ap-1255	67	6	take	take	VERB
ap-1255	67	7	values	value	NOUN
ap-1255	67	8	in	in	ADP
ap-1255	67	9	the	the	DET
ap-1255	67	10	set	set	NOUN
ap-1255	67	11	{	{	PUNCT
ap-1255	67	12	δi	δi	VERB
ap-1255	68	1	|	|	ADV
ap-1255	68	2	i	i	NOUN
ap-1255	68	3	=	=	NOUN
ap-1255	68	4	0	0	NUM
ap-1255	68	5	,	,	PUNCT
ap-1255	68	6	1	1	NUM
ap-1255	68	7	,	,	PUNCT
ap-1255	68	8	.	.	PUNCT
ap-1255	68	9	.	.	PUNCT
ap-1255	69	1	.	.	PUNCT
ap-1255	70	1	}	}	PUNCT
ap-1255	70	2	,	,	PUNCT
ap-1255	70	3	where	where	SCONJ
ap-1255	70	4	δi	δi	VERB
ap-1255	70	5	=	=	PUNCT
ap-1255	70	6	∞∑	∞∑	NUM
ap-1255	70	7	j=1	j=1	NOUN
ap-1255	70	8	ti+j	ti+j	NUM
ap-1255	70	9	βj	βj	PROPN
ap-1255	70	10	and	and	CCONJ
ap-1255	70	11	dβ(1	dβ(1	PROPN
ap-1255	70	12	)	)	PUNCT
ap-1255	70	13	=	=	SYM
ap-1255	71	1	t1t2	t1t2	PROPN
ap-1255	71	2	.	.	PUNCT
ap-1255	71	3	.	.	PUNCT
ap-1255	72	1	..	..	PUNCT
ap-1255	73	1	moreover	moreover	ADV
ap-1255	73	2	,	,	PUNCT
ap-1255	73	3	the	the	DET
ap-1255	73	4	sequence	sequence	NOUN
ap-1255	73	5	coding	code	VERB
ap-1255	73	6	the	the	DET
ap-1255	73	7	distances	distance	NOUN
ap-1255	73	8	in	in	ADP
ap-1255	73	9	zβ	zβ	PROPN
ap-1255	73	10	is	be	AUX
ap-1255	73	11	known	know	VERB
ap-1255	73	12	to	to	PART
ap-1255	73	13	be	be	AUX
ap-1255	73	14	invariant	invariant	ADJ
ap-1255	73	15	under	under	ADP
ap-1255	73	16	a	a	DET
ap-1255	73	17	substitution	substitution	NOUN
ap-1255	73	18	provided	provide	VERB
ap-1255	73	19	dβ(1	dβ(1	PROPN
ap-1255	73	20	)	)	PUNCT
ap-1255	73	21	is	be	AUX
ap-1255	73	22	eventually	eventually	ADV
ap-1255	73	23	periodic	periodic	ADJ
ap-1255	73	24	[	[	X
ap-1255	73	25	5	5	NUM
ap-1255	73	26	]	]	PUNCT
ap-1255	73	27	.	.	PUNCT
ap-1255	74	1	the	the	DET
ap-1255	74	2	form	form	NOUN
ap-1255	74	3	of	of	ADP
ap-1255	74	4	this	this	DET
ap-1255	74	5	substitution	substitution	NOUN
ap-1255	74	6	also	also	ADV
ap-1255	74	7	depends	depend	VERB
ap-1255	74	8	on	on	ADP
ap-1255	74	9	dβ(1	dβ(1	PROPN
ap-1255	74	10	)	)	PUNCT
ap-1255	74	11	.	.	PUNCT
ap-1255	75	1	if	if	SCONJ
ap-1255	75	2	we	we	PRON
ap-1255	75	3	consider	consider	VERB
ap-1255	75	4	β	β	PRON
ap-1255	75	5	an	an	DET
ap-1255	75	6	algebraic	algebraic	ADJ
ap-1255	75	7	integer	integer	NOUN
ap-1255	75	8	,	,	PUNCT
ap-1255	75	9	then	then	ADV
ap-1255	75	10	obviously	obviously	ADV
ap-1255	75	11	fin(β	fin(β	PROPN
ap-1255	75	12	)	)	PUNCT
ap-1255	76	1	⊂	⊂	PROPN
ap-1255	76	2	z[β−1]+	z[β−1]+	NOUN
ap-1255	76	3	.	.	PUNCT
ap-1255	77	1	the	the	DET
ap-1255	77	2	converse	converse	NOUN
ap-1255	77	3	inclusion	inclusion	NOUN
ap-1255	77	4	,	,	PUNCT
ap-1255	77	5	which	which	PRON
ap-1255	77	6	is	be	AUX
ap-1255	77	7	very	very	ADV
ap-1255	77	8	important	important	ADJ
ap-1255	77	9	for	for	ADP
ap-1255	77	10	the	the	DET
ap-1255	77	11	construction	construction	NOUN
ap-1255	77	12	of	of	ADP
ap-1255	77	13	the	the	DET
ap-1255	77	14	tiling	tiling	NOUN
ap-1255	77	15	and	and	CCONJ
ap-1255	77	16	also	also	ADV
ap-1255	77	17	for	for	ADP
ap-1255	77	18	the	the	DET
ap-1255	77	19	arithmetical	arithmetical	ADJ
ap-1255	77	20	properties	property	NOUN
ap-1255	77	21	of	of	ADP
ap-1255	77	22	the	the	DET
ap-1255	77	23	system	system	NOUN
ap-1255	77	24	,	,	PUNCT
ap-1255	77	25	does	do	AUX
ap-1255	77	26	not	not	PART
ap-1255	77	27	hold	hold	VERB
ap-1255	77	28	in	in	ADP
ap-1255	77	29	general	general	ADJ
ap-1255	77	30	.	.	PUNCT
ap-1255	78	1	an	an	DET
ap-1255	78	2	algebraic	algebraic	ADJ
ap-1255	78	3	integer	integer	NOUN
ap-1255	78	4	β	β	X
ap-1255	78	5	for	for	ADP
ap-1255	78	6	which	which	PRON
ap-1255	78	7	fin(β	fin(β	PROPN
ap-1255	78	8	)	)	PUNCT
ap-1255	78	9	=	=	NOUN
ap-1255	78	10	z[β−1]+	z[β−1]+	NOUN
ap-1255	78	11	holds	hold	NOUN
ap-1255	78	12	,	,	PUNCT
ap-1255	78	13	is	be	AUX
ap-1255	78	14	said	say	VERB
ap-1255	78	15	to	to	PART
ap-1255	78	16	have	have	VERB
ap-1255	78	17	property	property	NOUN
ap-1255	78	18	(	(	PUNCT
ap-1255	78	19	f	f	NOUN
ap-1255	78	20	)	)	PUNCT
ap-1255	78	21	.	.	PUNCT
ap-1255	79	1	3	3	NUM
ap-1255	79	2	ito	ito	PROPN
ap-1255	79	3	-	-	NOUN
ap-1255	79	4	sadahiro	sadahiro	PROPN
ap-1255	79	5	(	(	PUNCT
ap-1255	79	6	−β)-expansions	−β)-expansion	NOUN
ap-1255	79	7	now	now	ADV
ap-1255	79	8	consider	consider	VERB
ap-1255	79	9	the	the	DET
ap-1255	79	10	real	real	ADJ
ap-1255	79	11	base	base	NOUN
ap-1255	79	12	−β	−β	NOUN
ap-1255	79	13	<	<	X
ap-1255	79	14	−1	−1	NOUN
ap-1255	79	15	and	and	CCONJ
ap-1255	79	16	the	the	DET
ap-1255	79	17	transformation	transformation	NOUN
ap-1255	79	18	t−β	t−β	NOUN
ap-1255	79	19	:	:	PUNCT
ap-1255	79	20	[	[	PUNCT
ap-1255	79	21	−β	−β	NOUN
ap-1255	79	22	β	β	X
ap-1255	79	23	+	+	CCONJ
ap-1255	79	24	1	1	NUM
ap-1255	79	25	,	,	PUNCT
ap-1255	79	26	1	1	NUM
ap-1255	79	27	β	β	X
ap-1255	79	28	+	+	NOUN
ap-1255	79	29	1	1	NUM
ap-1255	79	30	)	)	PUNCT
ap-1255	79	31	→	→	PUNCT
ap-1255	79	32	[	[	PUNCT
ap-1255	79	33	−β	−β	NOUN
ap-1255	79	34	β	β	X
ap-1255	79	35	+	+	CCONJ
ap-1255	79	36	1	1	NUM
ap-1255	79	37	,	,	PUNCT
ap-1255	79	38	1	1	NUM
ap-1255	79	39	β	β	X
ap-1255	79	40	+	+	NOUN
ap-1255	79	41	1	1	X
ap-1255	79	42	)	)	PUNCT
ap-1255	79	43	defined	define	VERB
ap-1255	79	44	by	by	ADP
ap-1255	79	45	the	the	DET
ap-1255	79	46	prescription	prescription	NOUN
ap-1255	79	47	t−β(x	t−β(x	NOUN
ap-1255	79	48	)	)	PUNCT
ap-1255	80	1	=	=	SYM
ap-1255	80	2	−βx	−βx	NOUN
ap-1255	80	3	−	−	NOUN
ap-1255	80	4	⌊	⌊	NOUN
ap-1255	80	5	−	−	NOUN
ap-1255	80	6	βx+	βx+	NOUN
ap-1255	80	7	β	β	X
ap-1255	80	8	β	β	X
ap-1255	80	9	+	+	CCONJ
ap-1255	80	10	1	1	NUM
ap-1255	80	11	⌋	⌋	NOUN
ap-1255	80	12	.	.	PUNCT
ap-1255	81	1	every	every	DET
ap-1255	81	2	number	number	NOUN
ap-1255	81	3	x	x	SYM
ap-1255	81	4	∈	∈	PROPN
ap-1255	81	5	[	[	PUNCT
ap-1255	81	6	−β	−β	NOUN
ap-1255	81	7	β	β	X
ap-1255	81	8	+	+	CCONJ
ap-1255	81	9	1	1	NUM
ap-1255	81	10	,	,	PUNCT
ap-1255	81	11	1	1	NUM
ap-1255	81	12	β	β	X
ap-1255	81	13	+	+	NOUN
ap-1255	81	14	1	1	NUM
ap-1255	81	15	)	)	PUNCT
ap-1255	81	16	can	can	AUX
ap-1255	81	17	be	be	AUX
ap-1255	81	18	represented	represent	VERB
ap-1255	81	19	in	in	ADP
ap-1255	81	20	the	the	DET
ap-1255	81	21	form	form	NOUN
ap-1255	81	22	x	x	PUNCT
ap-1255	82	1	=	=	SYM
ap-1255	82	2	x1	x1	NUM
ap-1255	82	3	−β	−β	NOUN
ap-1255	82	4	+	+	CCONJ
ap-1255	82	5	x2	x2	PROPN
ap-1255	82	6	(	(	PUNCT
ap-1255	82	7	−β)2	−β)2	NOUN
ap-1255	82	8	+	+	SYM
ap-1255	82	9	x3	x3	ADJ
ap-1255	82	10	(	(	PUNCT
ap-1255	82	11	−β)3	−β)3	PROPN
ap-1255	82	12	+	+	X
ap-1255	82	13	·	·	PUNCT
ap-1255	82	14	·	·	PUNCT
ap-1255	82	15	·	·	PUNCT
ap-1255	82	16	,	,	PUNCT
ap-1255	82	17	where	where	SCONJ
ap-1255	82	18	xi	xi	X
ap-1255	82	19	=	=	PUNCT
ap-1255	82	20	⌊	⌊	VERB
ap-1255	82	21	−	−	NOUN
ap-1255	82	22	βt	βt	ADP
ap-1255	82	23	i−1	i−1	PROPN
ap-1255	82	24	−β	−β	PROPN
ap-1255	82	25	(	(	PUNCT
ap-1255	82	26	x	x	X
ap-1255	82	27	)	)	PUNCT
ap-1255	82	28	+	+	CCONJ
ap-1255	82	29	β	β	X
ap-1255	82	30	β	β	X
ap-1255	82	31	+	+	NOUN
ap-1255	82	32	1	1	NUM
ap-1255	82	33	⌋	⌋	NOUN
ap-1255	82	34	.	.	PUNCT
ap-1255	83	1	the	the	DET
ap-1255	83	2	representation	representation	NOUN
ap-1255	83	3	of	of	ADP
ap-1255	83	4	x	x	PUNCT
ap-1255	83	5	in	in	ADP
ap-1255	83	6	such	such	DET
ap-1255	83	7	a	a	DET
ap-1255	83	8	form	form	NOUN
ap-1255	83	9	is	be	AUX
ap-1255	83	10	called	call	VERB
ap-1255	83	11	the	the	DET
ap-1255	83	12	(	(	PUNCT
ap-1255	83	13	−β)-expansion	−β)-expansion	NOUN
ap-1255	83	14	of	of	ADP
ap-1255	83	15	x	x	PUNCT
ap-1255	83	16	and	and	CCONJ
ap-1255	83	17	is	be	AUX
ap-1255	83	18	denoted	denote	VERB
ap-1255	83	19	d−β(x	d−β(x	NOUN
ap-1255	83	20	)	)	PUNCT
ap-1255	83	21	=	=	PUNCT
ap-1255	83	22	•x1x2x3	•x1x2x3	PROPN
ap-1255	83	23	.	.	PUNCT
ap-1255	83	24	.	.	PUNCT
ap-1255	83	25	.	.	PUNCT
ap-1255	84	1	8	8	NUM
ap-1255	84	2	acta	acta	PROPN
ap-1255	84	3	polytechnica	polytechnica	PROPN
ap-1255	84	4	vol	vol	NOUN
ap-1255	84	5	.	.	PROPN
ap-1255	85	1	50	50	NUM
ap-1255	85	2	no	no	NOUN
ap-1255	85	3	.	.	PUNCT
ap-1255	86	1	5/2010	5/2010	NUM
ap-1255	86	2	by	by	ADP
ap-1255	86	3	analogy	analogy	NOUN
ap-1255	86	4	to	to	ADP
ap-1255	86	5	the	the	DET
ap-1255	86	6	case	case	NOUN
ap-1255	86	7	of	of	ADP
ap-1255	86	8	rényi	rényi	PROPN
ap-1255	86	9	β	β	NOUN
ap-1255	86	10	-	-	NOUN
ap-1255	86	11	expansions	expansion	NOUN
ap-1255	86	12	,	,	PUNCT
ap-1255	86	13	we	we	PRON
ap-1255	86	14	use	use	VERB
ap-1255	86	15	for	for	ADP
ap-1255	86	16	the	the	DET
ap-1255	86	17	(	(	PUNCT
ap-1255	86	18	−β)-expansion	−β)-expansion	NOUN
ap-1255	86	19	of	of	ADP
ap-1255	86	20	x	x	PUNCT
ap-1255	86	21	∈	∈	PROPN
ap-1255	86	22	r	r	NOUN
ap-1255	86	23	a	a	DET
ap-1255	86	24	suitable	suitable	ADJ
ap-1255	86	25	exponent	exponent	NOUN
ap-1255	86	26	l	l	PROPN
ap-1255	86	27	∈	∈	PROPN
ap-1255	86	28	n	n	PRON
ap-1255	86	29	such	such	ADJ
ap-1255	86	30	that	that	SCONJ
ap-1255	86	31	x	x	X
ap-1255	86	32	(	(	PUNCT
ap-1255	86	33	−β)l	−β)l	NOUN
ap-1255	86	34	∈	∈	PROPN
ap-1255	86	35	[	[	PUNCT
ap-1255	86	36	−β	−β	NOUN
ap-1255	86	37	β	β	X
ap-1255	86	38	+	+	CCONJ
ap-1255	86	39	1	1	NUM
ap-1255	86	40	,	,	PUNCT
ap-1255	86	41	1	1	NUM
ap-1255	86	42	β	β	X
ap-1255	86	43	+	+	NOUN
ap-1255	86	44	1	1	NUM
ap-1255	86	45	)	)	PUNCT
ap-1255	86	46	.	.	PUNCT
ap-1255	87	1	it	it	PRON
ap-1255	87	2	is	be	AUX
ap-1255	87	3	shown	show	VERB
ap-1255	87	4	easily	easily	ADV
ap-1255	87	5	that	that	SCONJ
ap-1255	87	6	the	the	DET
ap-1255	87	7	digits	digit	NOUN
ap-1255	87	8	xi	xi	ADP
ap-1255	87	9	of	of	ADP
ap-1255	87	10	a	a	DET
ap-1255	87	11	(	(	PUNCT
ap-1255	87	12	−β)-expansion	−β)-expansion	NOUN
ap-1255	87	13	belong	belong	VERB
ap-1255	87	14	to	to	ADP
ap-1255	87	15	the	the	DET
ap-1255	87	16	set	set	NOUN
ap-1255	87	17	{	{	PUNCT
ap-1255	87	18	0	0	NUM
ap-1255	87	19	,	,	PUNCT
ap-1255	87	20	1	1	NUM
ap-1255	87	21	,	,	PUNCT
ap-1255	87	22	.	.	PUNCT
ap-1255	87	23	.	.	PUNCT
ap-1255	88	1	.	.	PUNCT
ap-1255	89	1	,	,	PUNCT
ap-1255	89	2	�	�	PROPN
ap-1255	89	3	β	β	X
ap-1255	89	4	�	�	NOUN
ap-1255	89	5	}	}	PUNCT
ap-1255	89	6	.	.	PUNCT
ap-1255	90	1	in	in	ADP
ap-1255	90	2	order	order	NOUN
ap-1255	90	3	to	to	PART
ap-1255	90	4	describe	describe	VERB
ap-1255	90	5	strings	string	NOUN
ap-1255	90	6	that	that	PRON
ap-1255	90	7	arise	arise	VERB
ap-1255	90	8	as	as	ADP
ap-1255	90	9	(	(	PUNCT
ap-1255	90	10	−β)-expansions	−β)-expansion	NOUN
ap-1255	90	11	of	of	ADP
ap-1255	90	12	some	some	DET
ap-1255	90	13	x	x	SYM
ap-1255	90	14	∈	∈	PROPN
ap-1255	90	15	[	[	PUNCT
ap-1255	90	16	−β	−β	NOUN
ap-1255	90	17	β	β	X
ap-1255	90	18	+	+	CCONJ
ap-1255	90	19	1	1	NUM
ap-1255	90	20	,	,	PUNCT
ap-1255	90	21	1	1	NUM
ap-1255	90	22	β	β	X
ap-1255	90	23	+	+	NOUN
ap-1255	90	24	1	1	NUM
ap-1255	90	25	)	)	PUNCT
ap-1255	90	26	,	,	PUNCT
ap-1255	90	27	so	so	ADV
ap-1255	90	28	-	-	PUNCT
ap-1255	90	29	called	call	VERB
ap-1255	90	30	(	(	PUNCT
ap-1255	90	31	−β)admissible	−β)admissible	ADJ
ap-1255	90	32	digit	digit	NOUN
ap-1255	90	33	strings	string	NOUN
ap-1255	90	34	,	,	PUNCT
ap-1255	90	35	we	we	PRON
ap-1255	90	36	will	will	AUX
ap-1255	90	37	use	use	VERB
ap-1255	90	38	the	the	DET
ap-1255	90	39	notation	notation	NOUN
ap-1255	90	40	introduced	introduce	VERB
ap-1255	90	41	in	in	ADP
ap-1255	90	42	[	[	X
ap-1255	90	43	6	6	NUM
ap-1255	90	44	]	]	PUNCT
ap-1255	90	45	.	.	PUNCT
ap-1255	91	1	we	we	PRON
ap-1255	91	2	denote	denote	VERB
ap-1255	91	3	lβ	lβ	ADP
ap-1255	91	4	=	=	VERB
ap-1255	91	5	−β	−β	PROPN
ap-1255	91	6	β	β	X
ap-1255	92	1	+	+	CCONJ
ap-1255	92	2	1	1	NUM
ap-1255	92	3	and	and	CCONJ
ap-1255	92	4	rβ	rβ	VERB
ap-1255	93	1	=	=	SYM
ap-1255	93	2	1	1	NUM
ap-1255	93	3	β	β	X
ap-1255	93	4	+	+	NOUN
ap-1255	93	5	1	1	NUM
ap-1255	93	6	the	the	DET
ap-1255	93	7	left	left	ADJ
ap-1255	93	8	and	and	CCONJ
ap-1255	93	9	right	right	ADJ
ap-1255	93	10	end	end	NOUN
ap-1255	93	11	-	-	PUNCT
ap-1255	93	12	point	point	NOUN
ap-1255	93	13	of	of	ADP
ap-1255	93	14	the	the	DET
ap-1255	93	15	definition	definition	NOUN
ap-1255	93	16	interval	interval	NOUN
ap-1255	93	17	iβ	iβ	ADP
ap-1255	93	18	of	of	ADP
ap-1255	93	19	the	the	DET
ap-1255	93	20	transformation	transformation	NOUN
ap-1255	93	21	t−β	t−β	NOUN
ap-1255	93	22	,	,	PUNCT
ap-1255	93	23	respectively	respectively	ADV
ap-1255	93	24	.	.	PUNCT
ap-1255	94	1	that	that	PRON
ap-1255	94	2	is	be	AUX
ap-1255	94	3	iβ	iβ	ADP
ap-1255	94	4	=	=	PUNCT
ap-1255	95	1	[	[	X
ap-1255	95	2	lβ	lβ	INTJ
ap-1255	95	3	,	,	PUNCT
ap-1255	95	4	rβ	rβ	PROPN
ap-1255	95	5	)	)	PUNCT
ap-1255	95	6	.	.	PUNCT
ap-1255	96	1	we	we	PRON
ap-1255	96	2	also	also	ADV
ap-1255	96	3	denote	denote	VERB
ap-1255	96	4	d−β(lβ	d−β(lβ	NOUN
ap-1255	96	5	)	)	PUNCT
ap-1255	96	6	=	=	SYM
ap-1255	97	1	d1d2d3	d1d2d3	NOUN
ap-1255	97	2	.	.	PUNCT
ap-1255	97	3	.	.	PUNCT
ap-1255	97	4	.	.	PUNCT
ap-1255	98	1	theorem	theorem	ADJ
ap-1255	98	2	2	2	NUM
ap-1255	98	3	(	(	PUNCT
ap-1255	98	4	[	[	X
ap-1255	98	5	6	6	NUM
ap-1255	98	6	]	]	SYM
ap-1255	98	7	)	)	PUNCT
ap-1255	98	8	a	a	DET
ap-1255	98	9	string	string	NOUN
ap-1255	98	10	x1x2x3	x1x2x3	PROPN
ap-1255	98	11	·	·	PUNCT
ap-1255	98	12	·	·	PUNCT
ap-1255	98	13	·	·	PUNCT
ap-1255	99	1	over	over	ADP
ap-1255	99	2	the	the	DET
ap-1255	99	3	alphabet	alphabet	NOUN
ap-1255	99	4	{	{	PUNCT
ap-1255	99	5	0	0	NUM
ap-1255	99	6	,	,	PUNCT
ap-1255	99	7	1	1	NUM
ap-1255	99	8	,	,	PUNCT
ap-1255	99	9	.	.	PUNCT
ap-1255	99	10	.	.	PUNCT
ap-1255	99	11	.	.	PUNCT
ap-1255	100	1	,	,	PUNCT
ap-1255	100	2	�	�	PROPN
ap-1255	100	3	β	β	SYM
ap-1255	100	4	�	�	PROPN
ap-1255	100	5	}	}	PUNCT
ap-1255	100	6	is	be	AUX
ap-1255	100	7	(	(	PUNCT
ap-1255	100	8	−β)-admissible	−β)-admissible	ADJ
ap-1255	100	9	,	,	PUNCT
ap-1255	100	10	if	if	SCONJ
ap-1255	100	11	and	and	CCONJ
ap-1255	100	12	only	only	ADV
ap-1255	100	13	if	if	SCONJ
ap-1255	100	14	for	for	ADP
ap-1255	100	15	all	all	DET
ap-1255	100	16	i	i	PRON
ap-1255	100	17	=	=	NOUN
ap-1255	100	18	1	1	NUM
ap-1255	100	19	,	,	PUNCT
ap-1255	100	20	2	2	NUM
ap-1255	100	21	,	,	PUNCT
ap-1255	100	22	3	3	NUM
ap-1255	100	23	,	,	PUNCT
ap-1255	100	24	.	.	PUNCT
ap-1255	100	25	.	.	PUNCT
ap-1255	101	1	.	.	PUNCT
ap-1255	101	2	,	,	PUNCT
ap-1255	101	3	d−β(lβ	d−β(lβ	NOUN
ap-1255	101	4	)	)	PUNCT
ap-1255	101	5	�	�	PROPN
ap-1255	101	6	alt	alt	VERB
ap-1255	101	7	xixi+1xi+2	xixi+1xi+2	PROPN
ap-1255	101	8	≺alt	≺alt	PROPN
ap-1255	101	9	d∗−β(rβ	d∗−β(rβ	NOUN
ap-1255	101	10	)	)	PUNCT
ap-1255	101	11	,	,	PUNCT
ap-1255	101	12	where	where	SCONJ
ap-1255	101	13	d∗−β(rβ	d∗−β(rβ	NOUN
ap-1255	101	14	)	)	PUNCT
ap-1255	102	1	=	=	PROPN
ap-1255	102	2	lim	lim	PROPN
ap-1255	102	3	ε→0	ε→0	NOUN
ap-1255	102	4	+	+	CCONJ
ap-1255	102	5	d−β(rβ	d−β(rβ	PROPN
ap-1255	102	6	−	−	ADP
ap-1255	102	7	ε	ε	PROPN
ap-1255	102	8	)	)	PUNCT
ap-1255	102	9	and	and	CCONJ
ap-1255	102	10	�	�	PROPN
ap-1255	102	11	alt	alt	NOUN
ap-1255	102	12	is	be	AUX
ap-1255	102	13	the	the	DET
ap-1255	102	14	alternate	alternate	ADJ
ap-1255	102	15	order	order	NOUN
ap-1255	102	16	.	.	PUNCT
ap-1255	103	1	recall	recall	VERB
ap-1255	103	2	that	that	SCONJ
ap-1255	103	3	the	the	DET
ap-1255	103	4	alternate	alternate	ADJ
ap-1255	103	5	order	order	NOUN
ap-1255	103	6	is	be	AUX
ap-1255	103	7	defined	define	VERB
ap-1255	103	8	as	as	SCONJ
ap-1255	103	9	follows	follow	VERB
ap-1255	103	10	:	:	PUNCT
ap-1255	103	11	we	we	PRON
ap-1255	103	12	say	say	VERB
ap-1255	103	13	that	that	SCONJ
ap-1255	103	14	x1x2x3	x1x2x3	PROPN
ap-1255	103	15	.	.	PUNCT
ap-1255	103	16	.	.	PUNCT
ap-1255	103	17	.	.	PUNCT
ap-1255	104	1	≺alt	≺alt	NOUN
ap-1255	104	2	y1y2y3	y1y2y3	NOUN
ap-1255	104	3	.	.	PUNCT
ap-1255	104	4	.	.	PUNCT
ap-1255	105	1	.	.	PUNCT
ap-1255	106	1	,	,	PUNCT
ap-1255	106	2	if	if	SCONJ
ap-1255	106	3	(	(	PUNCT
ap-1255	106	4	−1)i(xi−yi	−1)i(xi−yi	NOUN
ap-1255	106	5	)	)	PUNCT
ap-1255	106	6	>	>	X
ap-1255	106	7	0	0	PUNCT
ap-1255	107	1	for	for	ADP
ap-1255	107	2	the	the	DET
ap-1255	107	3	smallest	small	ADJ
ap-1255	107	4	index	index	NOUN
ap-1255	107	5	i	i	PRON
ap-1255	107	6	satisfying	satisfy	VERB
ap-1255	107	7	xi	xi	X
ap-1255	107	8	=	=	SYM
ap-1255	107	9	yi	yi	PROPN
ap-1255	107	10	.	.	PUNCT
ap-1255	108	1	the	the	DET
ap-1255	108	2	relation	relation	NOUN
ap-1255	108	3	between	between	ADP
ap-1255	108	4	d∗−β(rβ	d∗−β(rβ	NOUN
ap-1255	108	5	)	)	PUNCT
ap-1255	108	6	and	and	CCONJ
ap-1255	108	7	d−β(lβ	d−β(lβ	NOUN
ap-1255	108	8	)	)	PUNCT
ap-1255	108	9	is	be	AUX
ap-1255	108	10	described	describe	VERB
ap-1255	108	11	in	in	ADP
ap-1255	108	12	the	the	DET
ap-1255	108	13	same	same	ADJ
ap-1255	108	14	paper	paper	NOUN
ap-1255	108	15	.	.	PUNCT
ap-1255	109	1	theorem	theorem	NOUN
ap-1255	109	2	3	3	NUM
ap-1255	109	3	(	(	PUNCT
ap-1255	109	4	[	[	X
ap-1255	109	5	6	6	NUM
ap-1255	109	6	]	]	PUNCT
ap-1255	109	7	)	)	PUNCT
ap-1255	109	8	let	let	VERB
ap-1255	109	9	d−β(lβ	d−β(lβ	NOUN
ap-1255	109	10	)	)	PUNCT
ap-1255	109	11	=	=	SYM
ap-1255	110	1	d1d2d3	d1d2d3	NOUN
ap-1255	110	2	.	.	PUNCT
ap-1255	110	3	.	.	PUNCT
ap-1255	110	4	.	.	PUNCT
ap-1255	111	1	if	if	SCONJ
ap-1255	111	2	d−β(lβ	d−β(lβ	NOUN
ap-1255	111	3	)	)	PUNCT
ap-1255	111	4	is	be	AUX
ap-1255	111	5	purely	purely	ADV
ap-1255	111	6	periodic	periodic	ADJ
ap-1255	111	7	with	with	ADP
ap-1255	111	8	odd	odd	ADJ
ap-1255	111	9	period	period	NOUN
ap-1255	111	10	-	-	PUNCT
ap-1255	111	11	length	length	NOUN
ap-1255	111	12	,	,	PUNCT
ap-1255	111	13	i.e.	i.e.	X
ap-1255	111	14	,	,	PUNCT
ap-1255	111	15	d−β(lβ	d−β(lβ	NOUN
ap-1255	111	16	)	)	PUNCT
ap-1255	111	17	=	=	PUNCT
ap-1255	111	18	(	(	PUNCT
ap-1255	111	19	d1d2	d1d2	X
ap-1255	111	20	·	·	PUNCT
ap-1255	111	21	·	·	PUNCT
ap-1255	111	22	·	·	PUNCT
ap-1255	111	23	d2l+1)ω	d2l+1)ω	NOUN
ap-1255	111	24	,	,	PUNCT
ap-1255	111	25	then	then	ADV
ap-1255	111	26	d∗−β(rβ	d∗−β(rβ	NOUN
ap-1255	111	27	)	)	PUNCT
ap-1255	111	28	=	=	SYM
ap-1255	111	29	(	(	PUNCT
ap-1255	111	30	0d1d2	0d1d2	X
ap-1255	111	31	·	·	PUNCT
ap-1255	111	32	·	·	PUNCT
ap-1255	111	33	·	·	PUNCT
ap-1255	111	34	d2l(d2l+1	d2l(d2l+1	NOUN
ap-1255	112	1	−	−	ADP
ap-1255	112	2	1))ω	1))ω	NUM
ap-1255	112	3	.	.	PUNCT
ap-1255	113	1	otherwise	otherwise	ADV
ap-1255	113	2	,	,	PUNCT
ap-1255	113	3	d∗−β(rβ	d∗−β(rβ	NOUN
ap-1255	113	4	)	)	PUNCT
ap-1255	113	5	=	=	SYM
ap-1255	113	6	0d−β(lβ	0d−β(lβ	NOUN
ap-1255	113	7	)	)	PUNCT
ap-1255	113	8	.	.	PUNCT
ap-1255	114	1	similarly	similarly	ADV
ap-1255	114	2	to	to	ADP
ap-1255	114	3	the	the	DET
ap-1255	114	4	rényi	rényi	PROPN
ap-1255	114	5	case	case	NOUN
ap-1255	114	6	,	,	PUNCT
ap-1255	114	7	one	one	PRON
ap-1255	114	8	can	can	AUX
ap-1255	114	9	define	define	VERB
ap-1255	114	10	the	the	DET
ap-1255	114	11	set	set	NOUN
ap-1255	114	12	of	of	ADP
ap-1255	114	13	(	(	PUNCT
ap-1255	114	14	−β)-integers	−β)-integer	NOUN
ap-1255	114	15	,	,	PUNCT
ap-1255	114	16	denoted	denote	VERB
ap-1255	114	17	z−β	z−β	PROPN
ap-1255	114	18	,	,	PUNCT
ap-1255	114	19	using	use	VERB
ap-1255	114	20	the	the	DET
ap-1255	114	21	admissible	admissible	ADJ
ap-1255	114	22	digit	digit	NOUN
ap-1255	114	23	strings	string	NOUN
ap-1255	114	24	.	.	PUNCT
ap-1255	115	1	z−β	z−β	NUM
ap-1255	115	2	:	:	PUNCT
ap-1255	115	3	=	=	X
ap-1255	115	4	{	{	PUNCT
ap-1255	115	5	ak(−β)k	ak(−β)k	NOUN
ap-1255	115	6	+	+	X
ap-1255	115	7	·	·	PUNCT
ap-1255	115	8	·	·	PUNCT
ap-1255	115	9	·	·	PUNCT
ap-1255	115	10	a1(−β	a1(−β	NUM
ap-1255	115	11	)	)	PUNCT
ap-1255	115	12	+	+	CCONJ
ap-1255	115	13	a0	a0	PROPN
ap-1255	115	14	|	|	PROPN
ap-1255	115	15	ak	ak	PROPN
ap-1255	115	16	·	·	PUNCT
ap-1255	115	17	·	·	PUNCT
ap-1255	115	18	·	·	PUNCT
ap-1255	116	1	a1a00ω	a1a00ω	NOUN
ap-1255	116	2	is	be	AUX
ap-1255	116	3	a	a	DET
ap-1255	116	4	(	(	PUNCT
ap-1255	116	5	−β)-admissible	−β)-admissible	ADJ
ap-1255	116	6	digit	digit	NOUN
ap-1255	116	7	string	string	NOUN
ap-1255	116	8	}	}	PUNCT
ap-1255	116	9	.	.	PUNCT
ap-1255	117	1	the	the	DET
ap-1255	117	2	set	set	NOUN
ap-1255	117	3	of	of	ADP
ap-1255	117	4	distances	distance	NOUN
ap-1255	117	5	between	between	ADP
ap-1255	117	6	consecutive	consecutive	ADJ
ap-1255	117	7	(	(	PUNCT
ap-1255	117	8	−β)-integers	−β)-integer	NOUN
ap-1255	117	9	has	have	AUX
ap-1255	117	10	been	be	AUX
ap-1255	117	11	described	describe	VERB
ap-1255	117	12	only	only	ADV
ap-1255	117	13	for	for	ADP
ap-1255	117	14	a	a	DET
ap-1255	117	15	particular	particular	ADJ
ap-1255	117	16	class	class	NOUN
ap-1255	117	17	of	of	ADP
ap-1255	117	18	β	β	NOUN
ap-1255	117	19	,	,	PUNCT
ap-1255	117	20	cf	cf	NOUN
ap-1255	117	21	.	.	PUNCT
ap-1255	118	1	[	[	X
ap-1255	118	2	3	3	NUM
ap-1255	118	3	]	]	PUNCT
ap-1255	118	4	.	.	PUNCT
ap-1255	119	1	4	4	NUM
ap-1255	119	2	balanced	balanced	ADJ
ap-1255	119	3	(	(	PUNCT
ap-1255	119	4	−β)-numeration	−β)-numeration	NOUN
ap-1255	119	5	system	system	NOUN
ap-1255	119	6	the	the	DET
ap-1255	119	7	last	last	ADJ
ap-1255	119	8	numeration	numeration	NOUN
ap-1255	119	9	system	system	NOUN
ap-1255	119	10	used	use	VERB
ap-1255	119	11	in	in	ADP
ap-1255	119	12	this	this	DET
ap-1255	119	13	paper	paper	NOUN
ap-1255	119	14	is	be	AUX
ap-1255	119	15	a	a	DET
ap-1255	119	16	slight	slight	ADJ
ap-1255	119	17	modification	modification	NOUN
ap-1255	119	18	of	of	ADP
ap-1255	119	19	(	(	PUNCT
ap-1255	119	20	−β)-numeration	−β)-numeration	NOUN
ap-1255	119	21	defined	define	VERB
ap-1255	119	22	by	by	ADP
ap-1255	119	23	ito	ito	PROPN
ap-1255	119	24	and	and	CCONJ
ap-1255	119	25	sadahiro	sadahiro	PROPN
ap-1255	119	26	.	.	PUNCT
ap-1255	120	1	let	let	VERB
ap-1255	120	2	−β	−β	PRON
ap-1255	120	3	<	<	PART
ap-1255	120	4	−1	−1	NOUN
ap-1255	120	5	be	be	AUX
ap-1255	120	6	the	the	DET
ap-1255	120	7	base	base	NOUN
ap-1255	120	8	and	and	CCONJ
ap-1255	120	9	consider	consider	VERB
ap-1255	120	10	the	the	DET
ap-1255	120	11	transformation	transformation	NOUN
ap-1255	120	12	s−β	s−β	PROPN
ap-1255	120	13	:	:	PUNCT
ap-1255	120	14	[	[	PUNCT
ap-1255	120	15	−1	−1	NOUN
ap-1255	120	16	2	2	NUM
ap-1255	120	17	,	,	PUNCT
ap-1255	120	18	1	1	NUM
ap-1255	120	19	2	2	NUM
ap-1255	120	20	)	)	PUNCT
ap-1255	120	21	→	→	PUNCT
ap-1255	120	22	[	[	PUNCT
ap-1255	120	23	−1	−1	NOUN
ap-1255	120	24	2	2	NUM
ap-1255	120	25	,	,	PUNCT
ap-1255	120	26	1	1	NUM
ap-1255	120	27	2	2	NUM
ap-1255	120	28	)	)	PUNCT
ap-1255	120	29	given	give	VERB
ap-1255	120	30	by	by	ADP
ap-1255	120	31	s−β	s−β	PROPN
ap-1255	120	32	=	=	PUNCT
ap-1255	120	33	−βx	−βx	NOUN
ap-1255	120	34	−	−	PROPN
ap-1255	120	35	⌊	⌊	NOUN
ap-1255	120	36	−βx+	−βx+	NOUN
ap-1255	120	37	1	1	NUM
ap-1255	120	38	2	2	NUM
ap-1255	120	39	⌋	⌋	NOUN
ap-1255	120	40	.	.	PUNCT
ap-1255	121	1	the	the	DET
ap-1255	121	2	balanced	balanced	ADJ
ap-1255	121	3	(	(	PUNCT
ap-1255	121	4	−β)-expansion	−β)-expansion	NOUN
ap-1255	121	5	of	of	ADP
ap-1255	121	6	a	a	DET
ap-1255	121	7	number	number	NOUN
ap-1255	121	8	x	x	SYM
ap-1255	121	9	∈	∈	PROPN
ap-1255	121	10	[	[	PUNCT
ap-1255	121	11	−1	−1	NOUN
ap-1255	121	12	2	2	NUM
ap-1255	121	13	,	,	PUNCT
ap-1255	121	14	1	1	NUM
ap-1255	121	15	2	2	NUM
ap-1255	121	16	)	)	PUNCT
ap-1255	121	17	,	,	PUNCT
ap-1255	121	18	denoted	denote	VERB
ap-1255	121	19	db,−β(x	db,−β(x	NOUN
ap-1255	121	20	)	)	PUNCT
ap-1255	121	21	=	=	SYM
ap-1255	121	22	•x1x2x3	•x1x2x3	PROPN
ap-1255	121	23	.	.	PUNCT
ap-1255	121	24	.	.	PUNCT
ap-1255	122	1	.	.	PUNCT
ap-1255	122	2	,	,	PUNCT
ap-1255	122	3	is	be	AUX
ap-1255	122	4	x	x	X
ap-1255	122	5	=	=	SYM
ap-1255	122	6	x1	x1	PRON
ap-1255	122	7	−β	−β	NOUN
ap-1255	123	1	+	+	CCONJ
ap-1255	123	2	x2	x2	PROPN
ap-1255	123	3	(	(	PUNCT
ap-1255	123	4	−β)2	−β)2	NOUN
ap-1255	123	5	+	+	SYM
ap-1255	123	6	x3	x3	ADJ
ap-1255	123	7	(	(	PUNCT
ap-1255	123	8	−β)3	−β)3	PROPN
ap-1255	123	9	+	+	X
ap-1255	123	10	.	.	PUNCT
ap-1255	123	11	.	.	PUNCT
ap-1255	123	12	.	.	PUNCT
ap-1255	124	1	,	,	PUNCT
ap-1255	124	2	where	where	SCONJ
ap-1255	124	3	xi	xi	X
ap-1255	124	4	=	=	PUNCT
ap-1255	124	5	⌊	⌊	PROPN
ap-1255	124	6	−βsi−1	−βsi−1	PROPN
ap-1255	124	7	−β	−β	NOUN
ap-1255	124	8	(	(	PUNCT
ap-1255	124	9	x	x	X
ap-1255	124	10	)	)	PUNCT
ap-1255	124	11	+	+	CCONJ
ap-1255	124	12	1	1	NUM
ap-1255	124	13	2	2	NUM
ap-1255	124	14	⌋	⌋	NOUN
ap-1255	124	15	.	.	PUNCT
ap-1255	125	1	also	also	ADV
ap-1255	125	2	in	in	ADP
ap-1255	125	3	this	this	DET
ap-1255	125	4	case	case	NOUN
ap-1255	125	5	we	we	PRON
ap-1255	125	6	use	use	VERB
ap-1255	125	7	for	for	ADP
ap-1255	125	8	the	the	DET
ap-1255	125	9	(	(	PUNCT
ap-1255	125	10	−β)-expansion	−β)-expansion	NOUN
ap-1255	125	11	of	of	ADP
ap-1255	125	12	x	x	PUNCT
ap-1255	125	13	∈	∈	PROPN
ap-1255	125	14	r	r	NOUN
ap-1255	125	15	a	a	DET
ap-1255	125	16	suitable	suitable	ADJ
ap-1255	125	17	exponent	exponent	NOUN
ap-1255	125	18	l	l	PROPN
ap-1255	125	19	∈	∈	PROPN
ap-1255	125	20	n	n	PRON
ap-1255	125	21	such	such	ADJ
ap-1255	125	22	that	that	SCONJ
ap-1255	125	23	x	x	X
ap-1255	125	24	(	(	PUNCT
ap-1255	125	25	−β)l	−β)l	PROPN
ap-1255	125	26	∈	∈	PROPN
ap-1255	125	27	[	[	PUNCT
ap-1255	125	28	−1	−1	NOUN
ap-1255	125	29	2	2	NUM
ap-1255	125	30	,	,	PUNCT
ap-1255	125	31	1	1	NUM
ap-1255	125	32	2	2	NUM
ap-1255	125	33	)	)	PUNCT
ap-1255	125	34	.	.	PUNCT
ap-1255	126	1	it	it	PRON
ap-1255	126	2	is	be	AUX
ap-1255	126	3	shown	show	VERB
ap-1255	126	4	easily	easily	ADV
ap-1255	126	5	that	that	SCONJ
ap-1255	126	6	the	the	DET
ap-1255	126	7	digits	digit	NOUN
ap-1255	126	8	xi	xi	ADP
ap-1255	126	9	of	of	ADP
ap-1255	126	10	a	a	DET
ap-1255	126	11	balanced	balanced	ADJ
ap-1255	126	12	(	(	PUNCT
ap-1255	126	13	−β)-expansion	−β)-expansion	NOUN
ap-1255	126	14	belong	belong	VERB
ap-1255	126	15	to	to	ADP
ap-1255	126	16	the	the	DET
ap-1255	126	17	set	set	NOUN
ap-1255	126	18	{	{	PUNCT
ap-1255	126	19	−	−	PROPN
ap-1255	126	20	⌊	⌊	PROPN
ap-1255	126	21	β	β	X
ap-1255	126	22	+	+	CCONJ
ap-1255	126	23	1	1	NUM
ap-1255	126	24	2	2	NUM
ap-1255	126	25	⌋	⌋	NOUN
ap-1255	126	26	,	,	PUNCT
ap-1255	126	27	.	.	PUNCT
ap-1255	126	28	.	.	PUNCT
ap-1255	126	29	.	.	PUNCT
ap-1255	127	1	,	,	PUNCT
ap-1255	127	2	⌊	⌊	VERB
ap-1255	127	3	β	β	X
ap-1255	128	1	+	+	CCONJ
ap-1255	128	2	1	1	NUM
ap-1255	128	3	2	2	NUM
ap-1255	128	4	⌋	⌋	NOUN
ap-1255	128	5	}	}	PUNCT
ap-1255	128	6	.	.	PUNCT
ap-1255	129	1	note	note	VERB
ap-1255	129	2	that	that	SCONJ
ap-1255	129	3	sometimes	sometimes	ADV
ap-1255	129	4	d	d	NOUN
ap-1255	129	5	is	be	AUX
ap-1255	129	6	used	use	VERB
ap-1255	129	7	instead	instead	ADV
ap-1255	129	8	of	of	ADP
ap-1255	129	9	−d	−d	PROPN
ap-1255	129	10	.	.	PUNCT
ap-1255	130	1	a	a	DET
ap-1255	130	2	digit	digit	NOUN
ap-1255	130	3	string	string	NOUN
ap-1255	130	4	x1x2x3	x1x2x3	PROPN
ap-1255	130	5	.	.	PUNCT
ap-1255	130	6	.	.	PUNCT
ap-1255	130	7	.	.	PUNCT
ap-1255	131	1	is	be	AUX
ap-1255	131	2	called	call	VERB
ap-1255	131	3	balanced	balanced	ADJ
ap-1255	131	4	(	(	PUNCT
ap-1255	131	5	−β)-admissible	−β)-admissible	ADJ
ap-1255	131	6	if	if	SCONJ
ap-1255	131	7	it	it	PRON
ap-1255	131	8	arises	arise	VERB
ap-1255	131	9	as	as	ADP
ap-1255	131	10	the	the	DET
ap-1255	131	11	balanced	balanced	ADJ
ap-1255	131	12	(	(	PUNCT
ap-1255	131	13	−β)-expansion	−β)-expansion	NOUN
ap-1255	131	14	of	of	ADP
ap-1255	131	15	some	some	DET
ap-1255	131	16	x	x	SYM
ap-1255	131	17	∈	∈	PROPN
ap-1255	131	18	[	[	PUNCT
ap-1255	131	19	−1	−1	NOUN
ap-1255	131	20	2	2	NUM
ap-1255	131	21	,	,	PUNCT
ap-1255	131	22	1	1	NUM
ap-1255	131	23	2	2	NUM
ap-1255	131	24	)	)	PUNCT
ap-1255	131	25	.	.	PUNCT
ap-1255	132	1	the	the	DET
ap-1255	132	2	two	two	NUM
ap-1255	132	3	following	follow	VERB
ap-1255	132	4	theorems	theorem	NOUN
ap-1255	132	5	by	by	ADP
ap-1255	132	6	dombek	dombek	NOUN
ap-1255	132	7	[	[	X
ap-1255	132	8	4	4	NUM
ap-1255	132	9	]	]	PUNCT
ap-1255	132	10	prove	prove	VERB
ap-1255	132	11	that	that	SCONJ
ap-1255	132	12	also	also	ADV
ap-1255	132	13	in	in	ADP
ap-1255	132	14	this	this	DET
ap-1255	132	15	case	case	NOUN
ap-1255	132	16	the	the	DET
ap-1255	132	17	admissible	admissible	ADJ
ap-1255	132	18	strings	string	NOUN
ap-1255	132	19	are	be	AUX
ap-1255	132	20	characterized	characterize	VERB
ap-1255	132	21	by	by	ADP
ap-1255	132	22	the	the	DET
ap-1255	132	23	balanced	balanced	ADJ
ap-1255	132	24	(	(	PUNCT
ap-1255	132	25	−β)-expansions	−β)-expansion	NOUN
ap-1255	132	26	of	of	ADP
ap-1255	132	27	the	the	DET
ap-1255	132	28	endpoints	endpoint	NOUN
ap-1255	132	29	of	of	ADP
ap-1255	132	30	the	the	DET
ap-1255	132	31	interval	interval	NOUN
ap-1255	132	32	[	[	PUNCT
ap-1255	132	33	−1	−1	NOUN
ap-1255	132	34	2	2	NUM
ap-1255	132	35	,	,	PUNCT
ap-1255	132	36	1	1	NUM
ap-1255	132	37	2	2	NUM
ap-1255	132	38	)	)	PUNCT
ap-1255	132	39	.	.	PUNCT
ap-1255	133	1	9	9	NUM
ap-1255	133	2	acta	acta	PROPN
ap-1255	133	3	polytechnica	polytechnica	PROPN
ap-1255	133	4	vol	vol	NOUN
ap-1255	133	5	.	.	PROPN
ap-1255	134	1	50	50	NUM
ap-1255	134	2	no	no	NOUN
ap-1255	134	3	.	.	PUNCT
ap-1255	135	1	5/2010	5/2010	NUM
ap-1255	135	2	theorem	theorem	NOUN
ap-1255	135	3	4	4	NUM
ap-1255	135	4	(	(	PUNCT
ap-1255	135	5	[	[	X
ap-1255	135	6	4	4	NUM
ap-1255	135	7	]	]	PUNCT
ap-1255	135	8	)	)	PUNCT
ap-1255	135	9	a	a	DET
ap-1255	135	10	string	string	NOUN
ap-1255	135	11	x1x2x3	x1x2x3	PROPN
ap-1255	135	12	.	.	PUNCT
ap-1255	135	13	.	.	PUNCT
ap-1255	135	14	.	.	PUNCT
ap-1255	136	1	over	over	ADP
ap-1255	136	2	the	the	DET
ap-1255	136	3	alphabet	alphabet	NOUN
ap-1255	136	4	{	{	PUNCT
ap-1255	136	5	−	−	PROPN
ap-1255	136	6	⌊	⌊	VERB
ap-1255	136	7	β	β	NOUN
ap-1255	137	1	+	+	CCONJ
ap-1255	137	2	1	1	NUM
ap-1255	137	3	2	2	NUM
ap-1255	137	4	⌋	⌋	NOUN
ap-1255	137	5	,	,	PUNCT
ap-1255	137	6	.	.	PUNCT
ap-1255	137	7	.	.	PUNCT
ap-1255	137	8	.	.	PUNCT
ap-1255	138	1	,	,	PUNCT
ap-1255	138	2	⌊	⌊	VERB
ap-1255	138	3	β	β	X
ap-1255	138	4	+	+	CCONJ
ap-1255	138	5	1	1	NUM
ap-1255	138	6	2	2	NUM
ap-1255	138	7	⌋	⌋	NOUN
ap-1255	138	8	}	}	PUNCT
ap-1255	138	9	is	be	AUX
ap-1255	138	10	balanced	balance	VERB
ap-1255	138	11	(	(	PUNCT
ap-1255	138	12	−β)-admissible	−β)-admissible	ADJ
ap-1255	138	13	if	if	SCONJ
ap-1255	138	14	and	and	CCONJ
ap-1255	138	15	only	only	ADV
ap-1255	138	16	if	if	SCONJ
ap-1255	138	17	for	for	ADP
ap-1255	138	18	all	all	DET
ap-1255	138	19	i	i	PRON
ap-1255	138	20	=	=	NOUN
ap-1255	138	21	1	1	NUM
ap-1255	138	22	,	,	PUNCT
ap-1255	138	23	2	2	NUM
ap-1255	138	24	,	,	PUNCT
ap-1255	138	25	3	3	NUM
ap-1255	138	26	,	,	PUNCT
ap-1255	138	27	.	.	PUNCT
ap-1255	138	28	.	.	PUNCT
ap-1255	138	29	.	.	PUNCT
ap-1255	139	1	db,−β	db,−β	NOUN
ap-1255	139	2	(	(	PUNCT
ap-1255	139	3	−1	−1	NOUN
ap-1255	139	4	2	2	X
ap-1255	139	5	)	)	PUNCT
ap-1255	139	6	�	�	PROPN
ap-1255	139	7	alt	alt	ADJ
ap-1255	139	8	xixi+1xi+2	xixi+1xi+2	PROPN
ap-1255	139	9	.	.	PUNCT
ap-1255	139	10	.	.	PUNCT
ap-1255	139	11	.	.	PUNCT
ap-1255	140	1	≺alt	≺alt	NOUN
ap-1255	140	2	d	d	PROPN
ap-1255	140	3	∗	∗	X
ap-1255	140	4	b,−β	b,−β	NOUN
ap-1255	140	5	(	(	PUNCT
ap-1255	140	6	1	1	NUM
ap-1255	140	7	2	2	NUM
ap-1255	140	8	)	)	PUNCT
ap-1255	140	9	,	,	PUNCT
ap-1255	140	10	where	where	SCONJ
ap-1255	140	11	d∗b,−β	d∗b,−β	NOUN
ap-1255	140	12	(	(	PUNCT
ap-1255	140	13	1	1	NUM
ap-1255	140	14	2	2	NUM
ap-1255	140	15	)	)	PUNCT
ap-1255	140	16	=	=	SYM
ap-1255	140	17	lim	lim	PROPN
ap-1255	140	18	ε→0	ε→0	NOUN
ap-1255	140	19	+	+	CCONJ
ap-1255	140	20	db,−β	db,−β	NOUN
ap-1255	140	21	(	(	PUNCT
ap-1255	140	22	1	1	NUM
ap-1255	140	23	2	2	NUM
ap-1255	140	24	−	−	PROPN
ap-1255	140	25	ε	ε	PROPN
ap-1255	140	26	)	)	PUNCT
ap-1255	140	27	.	.	PUNCT
ap-1255	141	1	theorem	theorem	ADJ
ap-1255	141	2	5	5	NUM
ap-1255	141	3	(	(	PUNCT
ap-1255	141	4	[	[	X
ap-1255	141	5	4	4	NUM
ap-1255	141	6	]	]	PUNCT
ap-1255	141	7	)	)	PUNCT
ap-1255	141	8	let	let	VERB
ap-1255	141	9	db,−β	db,−β	NOUN
ap-1255	141	10	(	(	PUNCT
ap-1255	141	11	−1	−1	NOUN
ap-1255	141	12	2	2	NUM
ap-1255	141	13	)	)	PUNCT
ap-1255	142	1	=	=	NOUN
ap-1255	143	1	d1d2d3	d1d2d3	NOUN
ap-1255	143	2	.	.	PUNCT
ap-1255	143	3	.	.	PUNCT
ap-1255	143	4	.	.	PUNCT
ap-1255	144	1	then	then	ADV
ap-1255	144	2	d∗b,−β	d∗b,−β	PROPN
ap-1255	144	3	(	(	PUNCT
ap-1255	144	4	1	1	NUM
ap-1255	144	5	2	2	NUM
ap-1255	144	6	)	)	PUNCT
ap-1255	144	7	=	=	VERB
ap-1255	144	8	⎧⎪⎨⎪⎩	⎧⎪⎨⎪⎩	PROPN
ap-1255	144	9	(	(	PUNCT
ap-1255	144	10	d1	d1	PROPN
ap-1255	144	11	.	.	PUNCT
ap-1255	144	12	.	.	PUNCT
ap-1255	144	13	.	.	PUNCT
ap-1255	145	1	d2l	d2l	PROPN
ap-1255	145	2	(	(	PUNCT
ap-1255	145	3	d2l+1	d2l+1	NOUN
ap-1255	145	4	−	−	PROPN
ap-1255	145	5	1	1	X
ap-1255	145	6	)	)	PUNCT
ap-1255	145	7	d1	d1	PROPN
ap-1255	145	8	.	.	PUNCT
ap-1255	145	9	.	.	PUNCT
ap-1255	145	10	.	.	PUNCT
ap-1255	146	1	d2l(d2l+1	d2l(d2l+1	PROPN
ap-1255	147	1	−	−	PROPN
ap-1255	147	2	1	1	NUM
ap-1255	147	3	)	)	PUNCT
ap-1255	147	4	)	)	PUNCT
ap-1255	148	1	ω	ω	PROPN
ap-1255	148	2	if	if	SCONJ
ap-1255	148	3	db,−β	db,−β	PROPN
ap-1255	148	4	(	(	PUNCT
ap-1255	148	5	−1	−1	NOUN
ap-1255	148	6	2	2	NUM
ap-1255	148	7	)	)	PUNCT
ap-1255	148	8	=	=	SYM
ap-1255	148	9	(	(	PUNCT
ap-1255	148	10	d1	d1	PROPN
ap-1255	148	11	.	.	PUNCT
ap-1255	148	12	.	.	PUNCT
ap-1255	148	13	.	.	PUNCT
ap-1255	149	1	d2l+1)ω	d2l+1)ω	NOUN
ap-1255	149	2	,	,	PUNCT
ap-1255	149	3	d1	d1	PROPN
ap-1255	149	4	d2	d2	PROPN
ap-1255	149	5	d3	d3	PROPN
ap-1255	149	6	.	.	PUNCT
ap-1255	149	7	.	.	PUNCT
ap-1255	149	8	.	.	PUNCT
ap-1255	150	1	otherwise	otherwise	ADV
ap-1255	150	2	.	.	PUNCT
ap-1255	151	1	the	the	DET
ap-1255	151	2	set	set	NOUN
ap-1255	151	3	of	of	ADP
ap-1255	151	4	balanced	balanced	ADJ
ap-1255	151	5	(	(	PUNCT
ap-1255	151	6	−β)-integers	−β)-integer	NOUN
ap-1255	151	7	,	,	PUNCT
ap-1255	151	8	denoted	denote	VERB
ap-1255	151	9	zb,−β	zb,−β	PROPN
ap-1255	151	10	,	,	PUNCT
ap-1255	151	11	is	be	AUX
ap-1255	151	12	defined	define	VERB
ap-1255	151	13	by	by	ADP
ap-1255	151	14	analogy	analogy	NOUN
ap-1255	151	15	to	to	ADP
ap-1255	151	16	the	the	DET
ap-1255	151	17	two	two	NUM
ap-1255	151	18	previous	previous	ADJ
ap-1255	151	19	cases	case	NOUN
ap-1255	151	20	.	.	PUNCT
ap-1255	152	1	zb,−β	zb,−β	NOUN
ap-1255	152	2	:	:	PUNCT
ap-1255	152	3	=	=	X
ap-1255	152	4	{	{	PUNCT
ap-1255	152	5	ak(−β)k	ak(−β)k	NOUN
ap-1255	152	6	+	+	CCONJ
ap-1255	152	7	.	.	PUNCT
ap-1255	152	8	.	.	PUNCT
ap-1255	152	9	.	.	PUNCT
ap-1255	153	1	a1(−β	a1(−β	VERB
ap-1255	153	2	)	)	PUNCT
ap-1255	154	1	+	+	CCONJ
ap-1255	154	2	a0	a0	PROPN
ap-1255	154	3	|	|	PROPN
ap-1255	154	4	ak	ak	PROPN
ap-1255	154	5	.	.	PUNCT
ap-1255	154	6	.	.	PUNCT
ap-1255	154	7	.	.	PUNCT
ap-1255	155	1	a1a00ω	a1a00ω	NOUN
ap-1255	155	2	is	be	AUX
ap-1255	155	3	a	a	DET
ap-1255	155	4	balanced	balanced	ADJ
ap-1255	155	5	(	(	PUNCT
ap-1255	155	6	−β)-admissible	−β)-admissible	ADJ
ap-1255	155	7	string	string	NOUN
ap-1255	155	8	}	}	PUNCT
ap-1255	155	9	.	.	PUNCT
ap-1255	156	1	5	5	NUM
ap-1255	156	2	constructing	constructing	NOUN
ap-1255	156	3	of	of	ADP
ap-1255	156	4	the	the	DET
ap-1255	156	5	tiling	tile	VERB
ap-1255	156	6	recall	recall	NOUN
ap-1255	156	7	that	that	SCONJ
ap-1255	156	8	a	a	DET
ap-1255	156	9	pisot	pisot	ADJ
ap-1255	156	10	number	number	NOUN
ap-1255	156	11	is	be	AUX
ap-1255	156	12	an	an	DET
ap-1255	156	13	algebraic	algebraic	ADJ
ap-1255	156	14	integer	integer	NOUN
ap-1255	156	15	such	such	ADJ
ap-1255	156	16	that	that	SCONJ
ap-1255	156	17	all	all	PRON
ap-1255	156	18	its	its	PRON
ap-1255	156	19	algebraic	algebraic	ADJ
ap-1255	156	20	conjugates	conjugate	NOUN
ap-1255	156	21	are	be	AUX
ap-1255	156	22	in	in	ADP
ap-1255	156	23	modulus	modulus	NOUN
ap-1255	156	24	strictly	strictly	ADV
ap-1255	156	25	smaller	small	ADJ
ap-1255	156	26	than	than	ADP
ap-1255	156	27	one	one	NUM
ap-1255	156	28	.	.	PUNCT
ap-1255	157	1	let	let	VERB
ap-1255	157	2	β	β	PRON
ap-1255	157	3	>	>	X
ap-1255	157	4	1	1	NUM
ap-1255	157	5	be	be	AUX
ap-1255	157	6	a	a	DET
ap-1255	157	7	pisot	pisot	ADJ
ap-1255	157	8	number	number	NOUN
ap-1255	157	9	of	of	ADP
ap-1255	157	10	degree	degree	NOUN
ap-1255	158	1	d	d	NOUN
ap-1255	158	2	=	=	SYM
ap-1255	158	3	r	r	NOUN
ap-1255	158	4	+	+	NOUN
ap-1255	159	1	2s	2s	X
ap-1255	159	2	.	.	PUNCT
ap-1255	160	1	we	we	PRON
ap-1255	160	2	denote	denote	VERB
ap-1255	160	3	β	β	X
ap-1255	160	4	=	=	SYM
ap-1255	160	5	β(1	β(1	PROPN
ap-1255	160	6	)	)	PUNCT
ap-1255	160	7	and	and	CCONJ
ap-1255	160	8	we	we	PRON
ap-1255	160	9	assume	assume	VERB
ap-1255	160	10	that	that	SCONJ
ap-1255	160	11	β(2	β(2	PROPN
ap-1255	160	12	)	)	PUNCT
ap-1255	160	13	,	,	PUNCT
ap-1255	160	14	.	.	PUNCT
ap-1255	160	15	.	.	PUNCT
ap-1255	160	16	.	.	PUNCT
ap-1255	161	1	,	,	PUNCT
ap-1255	161	2	β(r	β(r	NOUN
ap-1255	161	3	)	)	PUNCT
ap-1255	161	4	are	be	AUX
ap-1255	161	5	real	real	ADJ
ap-1255	161	6	conjugates	conjugate	NOUN
ap-1255	161	7	of	of	ADP
ap-1255	161	8	β	β	X
ap-1255	161	9	and	and	CCONJ
ap-1255	161	10	β(r+1	β(r+1	NUM
ap-1255	161	11	)	)	PUNCT
ap-1255	161	12	,	,	PUNCT
ap-1255	161	13	.	.	PUNCT
ap-1255	161	14	.	.	PUNCT
ap-1255	162	1	.	.	PUNCT
ap-1255	163	1	,	,	PUNCT
ap-1255	163	2	β(r+2s	β(r+2	NOUN
ap-1255	163	3	)	)	PUNCT
ap-1255	163	4	are	be	AUX
ap-1255	163	5	complex	complex	ADJ
ap-1255	163	6	conjugates	conjugate	NOUN
ap-1255	163	7	of	of	ADP
ap-1255	163	8	β	β	PRON
ap-1255	163	9	such	such	ADJ
ap-1255	163	10	that	that	SCONJ
ap-1255	163	11	β(r+j	β(r+j	NOUN
ap-1255	163	12	)	)	PUNCT
ap-1255	163	13	=	=	SYM
ap-1255	164	1	β(r+s+j	β(r+s+j	X
ap-1255	164	2	)	)	PUNCT
ap-1255	165	1	for	for	ADP
ap-1255	165	2	j	j	PROPN
ap-1255	165	3	=	=	SYM
ap-1255	165	4	1	1	PROPN
ap-1255	165	5	,	,	PUNCT
ap-1255	165	6	.	.	PUNCT
ap-1255	165	7	.	.	PUNCT
ap-1255	165	8	.	.	PUNCT
ap-1255	166	1	,	,	PUNCT
ap-1255	166	2	s.	s.	PROPN
ap-1255	166	3	denote	denote	VERB
ap-1255	166	4	by	by	ADP
ap-1255	166	5	x(j	x(j	PROPN
ap-1255	166	6	)	)	PUNCT
ap-1255	166	7	,	,	PUNCT
ap-1255	166	8	j	j	PROPN
ap-1255	166	9	=	=	SYM
ap-1255	166	10	1	1	NUM
ap-1255	166	11	,	,	PUNCT
ap-1255	166	12	.	.	PUNCT
ap-1255	166	13	.	.	PUNCT
ap-1255	167	1	.	.	PUNCT
ap-1255	168	1	,	,	PUNCT
ap-1255	168	2	n	n	CCONJ
ap-1255	168	3	the	the	DET
ap-1255	168	4	corresponding	corresponding	ADJ
ap-1255	168	5	conjugate	conjugate	NOUN
ap-1255	168	6	of	of	ADP
ap-1255	168	7	x	x	X
ap-1255	168	8	∈	∈	PROPN
ap-1255	168	9	q(β	q(β	PROPN
ap-1255	168	10	)	)	PUNCT
ap-1255	168	11	,	,	PUNCT
ap-1255	168	12	i.e.	i.e.	X
ap-1255	168	13	,	,	PUNCT
ap-1255	168	14	x	x	X
ap-1255	169	1	=	=	SYM
ap-1255	169	2	q0	q0	PROPN
ap-1255	169	3	+	+	CCONJ
ap-1255	169	4	q1β	q1β	PROPN
ap-1255	169	5	+	+	X
ap-1255	169	6	.	.	PUNCT
ap-1255	169	7	.	.	PUNCT
ap-1255	170	1	.+	.+	NOUN
ap-1255	170	2	qd−1β	qd−1β	NUM
ap-1255	171	1	q−1	q−1	PROPN
ap-1255	171	2	�	�	PROPN
ap-1255	171	3	→	→	SYM
ap-1255	171	4	x(j	x(j	PROPN
ap-1255	171	5	)	)	PUNCT
ap-1255	171	6	=	=	VERB
ap-1255	172	1	q0	q0	PROPN
ap-1255	172	2	+	+	CCONJ
ap-1255	172	3	q1β	q1β	PROPN
ap-1255	172	4	(	(	PUNCT
ap-1255	172	5	j	j	NOUN
ap-1255	172	6	)	)	PUNCT
ap-1255	172	7	+	+	CCONJ
ap-1255	172	8	.	.	PUNCT
ap-1255	172	9	.	.	PUNCT
ap-1255	173	1	.+	.+	NOUN
ap-1255	173	2	qd−1(β	qd−1(β	PROPN
ap-1255	173	3	(	(	PUNCT
ap-1255	173	4	j))q−1	j))q−1	PROPN
ap-1255	173	5	.	.	PUNCT
ap-1255	174	1	consider	consider	VERB
ap-1255	174	2	the	the	DET
ap-1255	174	3	map	map	NOUN
ap-1255	174	4	φ	φ	X
ap-1255	174	5	:	:	PUNCT
ap-1255	174	6	q(β)→	q(β)→	SYM
ap-1255	174	7	r	r	NOUN
ap-1255	174	8	d−1	d−1	PROPN
ap-1255	174	9	defined	define	VERB
ap-1255	174	10	by	by	ADP
ap-1255	174	11	φ(x	φ(x	NOUN
ap-1255	174	12	)	)	PUNCT
ap-1255	174	13	:	:	PUNCT
ap-1255	175	1	=	=	SYM
ap-1255	175	2	(	(	PUNCT
ap-1255	175	3	x(2	x(2	PROPN
ap-1255	175	4	)	)	PUNCT
ap-1255	175	5	,	,	PUNCT
ap-1255	175	6	.	.	PUNCT
ap-1255	175	7	.	.	PUNCT
ap-1255	175	8	.	.	PUNCT
ap-1255	176	1	,	,	PUNCT
ap-1255	176	2	x(r	x(r	PROPN
ap-1255	176	3	)	)	PUNCT
ap-1255	176	4	,	,	PUNCT
ap-1255	176	5	�	�	PROPN
ap-1255	176	6	(	(	PUNCT
ap-1255	176	7	x(r+1	x(r+1	PROPN
ap-1255	176	8	)	)	PUNCT
ap-1255	176	9	)	)	PUNCT
ap-1255	176	10	,	,	PUNCT
ap-1255	176	11	�	�	PROPN
ap-1255	176	12	(	(	PUNCT
ap-1255	176	13	x(r+1	x(r+1	PROPN
ap-1255	176	14	)	)	PUNCT
ap-1255	176	15	)	)	PUNCT
ap-1255	176	16	,	,	PUNCT
ap-1255	176	17	.	.	PUNCT
ap-1255	176	18	.	.	PUNCT
ap-1255	177	1	.	.	PUNCT
ap-1255	178	1	,	,	PUNCT
ap-1255	178	2	�	�	PROPN
ap-1255	178	3	(	(	PUNCT
ap-1255	178	4	x(r+s	x(r+s	PROPN
ap-1255	178	5	)	)	PUNCT
ap-1255	178	6	)	)	PUNCT
ap-1255	178	7	,	,	PUNCT
ap-1255	178	8	�	�	PROPN
ap-1255	178	9	(	(	PUNCT
ap-1255	178	10	x(r+s	x(r+s	PROPN
ap-1255	178	11	)	)	PUNCT
ap-1255	178	12	)	)	PUNCT
ap-1255	178	13	.	.	PUNCT
ap-1255	179	1	proposition	proposition	NOUN
ap-1255	179	2	6	6	NUM
ap-1255	179	3	(	(	PUNCT
ap-1255	179	4	[	[	X
ap-1255	179	5	1	1	NUM
ap-1255	179	6	]	]	PUNCT
ap-1255	179	7	)	)	PUNCT
ap-1255	179	8	let	let	VERB
ap-1255	179	9	β	β	PRON
ap-1255	179	10	>	>	X
ap-1255	179	11	1	1	NUM
ap-1255	179	12	be	be	AUX
ap-1255	179	13	a	a	DET
ap-1255	179	14	pisot	pisot	ADJ
ap-1255	179	15	number	number	NOUN
ap-1255	179	16	of	of	ADP
ap-1255	179	17	degree	degree	NOUN
ap-1255	179	18	d.	d.	PROPN
ap-1255	179	19	then	then	ADV
ap-1255	179	20	φ(z[β	φ(z[β	X
ap-1255	179	21	]	]	PUNCT
ap-1255	179	22	)	)	PUNCT
ap-1255	179	23	is	be	AUX
ap-1255	179	24	dense	dense	ADJ
ap-1255	179	25	in	in	ADP
ap-1255	179	26	rd−1	rd−1	PROPN
ap-1255	179	27	.	.	PUNCT
ap-1255	180	1	the	the	DET
ap-1255	180	2	map	map	NOUN
ap-1255	180	3	φ	φ	PROPN
ap-1255	180	4	is	be	AUX
ap-1255	180	5	used	use	VERB
ap-1255	180	6	to	to	PART
ap-1255	180	7	construct	construct	VERB
ap-1255	180	8	the	the	DET
ap-1255	180	9	tiling	tiling	NOUN
ap-1255	180	10	in	in	ADP
ap-1255	180	11	the	the	DET
ap-1255	180	12	following	following	ADJ
ap-1255	180	13	way	way	NOUN
ap-1255	180	14	.	.	PUNCT
ap-1255	181	1	let	let	VERB
ap-1255	181	2	w	w	NOUN
ap-1255	181	3	=	=	PROPN
ap-1255	181	4	w1	w1	NOUN
ap-1255	181	5	.	.	PUNCT
ap-1255	181	6	.	.	PUNCT
ap-1255	181	7	.	.	PUNCT
ap-1255	182	1	wl	wl	X
ap-1255	182	2	∈	∈	PROPN
ap-1255	182	3	{	{	PUNCT
ap-1255	182	4	0	0	NUM
ap-1255	182	5	,	,	PUNCT
ap-1255	182	6	1	1	NUM
ap-1255	182	7	,	,	PUNCT
ap-1255	182	8	.	.	PUNCT
ap-1255	182	9	.	.	PUNCT
ap-1255	182	10	.	.	PUNCT
ap-1255	183	1	,	,	PUNCT
ap-1255	183	2	�	�	PROPN
ap-1255	183	3	β	β	NOUN
ap-1255	183	4	�	�	PROPN
ap-1255	183	5	−	−	PROPN
ap-1255	183	6	1}∗	1}∗	PROPN
ap-1255	183	7	be	be	AUX
ap-1255	183	8	a	a	DET
ap-1255	183	9	finite	finite	ADJ
ap-1255	183	10	word	word	NOUN
ap-1255	183	11	such	such	ADJ
ap-1255	183	12	that	that	SCONJ
ap-1255	183	13	w0ω	w0ω	PROPN
ap-1255	183	14	is	be	AUX
ap-1255	183	15	an	an	DET
ap-1255	183	16	admissible	admissible	ADJ
ap-1255	183	17	digit	digit	NOUN
ap-1255	183	18	string	string	NOUN
ap-1255	183	19	.	.	PUNCT
ap-1255	184	1	we	we	PRON
ap-1255	184	2	define	define	VERB
ap-1255	184	3	the	the	DET
ap-1255	184	4	tile	tile	NOUN
ap-1255	184	5	tw	tw	NOUN
ap-1255	184	6	as	as	ADP
ap-1255	184	7	tw	tw	NOUN
ap-1255	184	8	:	:	PUNCT
ap-1255	184	9	=	=	SYM
ap-1255	184	10	{	{	PUNCT
ap-1255	184	11	φ(x	φ(x	NOUN
ap-1255	184	12	)	)	PUNCT
ap-1255	184	13	|	|	ADV
ap-1255	184	14	x	x	SYM
ap-1255	184	15	∈	∈	PROPN
ap-1255	184	16	fin(β	fin(β	PROPN
ap-1255	184	17	)	)	PUNCT
ap-1255	184	18	and	and	CCONJ
ap-1255	184	19	(	(	PUNCT
ap-1255	184	20	x)β	x)β	NOUN
ap-1255	184	21	=	=	PROPN
ap-1255	184	22	ak	ak	PROPN
ap-1255	184	23	.	.	PUNCT
ap-1255	184	24	.	.	PUNCT
ap-1255	184	25	.	.	PUNCT
ap-1255	185	1	x1x0•w1	x1x0•w1	X
ap-1255	185	2	·	·	PUNCT
ap-1255	185	3	·	·	PUNCT
ap-1255	185	4	·	·	PUNCT
ap-1255	185	5	wl	wl	X
ap-1255	185	6	}	}	PUNCT
ap-1255	185	7	.	.	PUNCT
ap-1255	186	1	the	the	DET
ap-1255	186	2	properties	property	NOUN
ap-1255	186	3	of	of	ADP
ap-1255	186	4	the	the	DET
ap-1255	186	5	tiling	tiling	NOUN
ap-1255	186	6	of	of	ADP
ap-1255	186	7	the	the	DET
ap-1255	186	8	euclidean	euclidean	ADJ
ap-1255	186	9	space	space	NOUN
ap-1255	186	10	using	use	VERB
ap-1255	186	11	tiles	tile	NOUN
ap-1255	186	12	tw	tw	NOUN
ap-1255	186	13	were	be	AUX
ap-1255	186	14	described	describe	VERB
ap-1255	186	15	by	by	ADP
ap-1255	186	16	akiyama	akiyama	PROPN
ap-1255	186	17	;	;	PUNCT
ap-1255	186	18	the	the	DET
ap-1255	186	19	results	result	NOUN
ap-1255	186	20	are	be	AUX
ap-1255	186	21	summarized	summarize	VERB
ap-1255	186	22	in	in	ADP
ap-1255	186	23	the	the	DET
ap-1255	186	24	following	follow	VERB
ap-1255	186	25	theorems	theorem	NOUN
ap-1255	186	26	.	.	PUNCT
ap-1255	187	1	theorem	theorem	NOUN
ap-1255	187	2	7	7	NUM
ap-1255	187	3	(	(	PUNCT
ap-1255	187	4	[	[	X
ap-1255	187	5	1	1	NUM
ap-1255	187	6	]	]	PUNCT
ap-1255	187	7	)	)	PUNCT
ap-1255	187	8	let	let	VERB
ap-1255	187	9	β	β	NOUN
ap-1255	187	10	be	be	AUX
ap-1255	187	11	a	a	DET
ap-1255	187	12	pisot	pisot	ADJ
ap-1255	187	13	unit	unit	NOUN
ap-1255	187	14	of	of	ADP
ap-1255	187	15	degree	degree	NOUN
ap-1255	187	16	d	d	NOUN
ap-1255	187	17	with	with	ADP
ap-1255	187	18	property	property	NOUN
ap-1255	187	19	(	(	PUNCT
ap-1255	187	20	f	f	NOUN
ap-1255	187	21	)	)	PUNCT
ap-1255	187	22	.	.	PUNCT
ap-1255	188	1	then	then	ADV
ap-1255	188	2	•	•	X
ap-1255	188	3	rd−1	rd−1	PROPN
ap-1255	188	4	=	=	SYM
ap-1255	188	5	⋃	⋃	ADP
ap-1255	188	6	w0ω	w0ω	ADJ
ap-1255	188	7	admissible	admissible	ADJ
ap-1255	188	8	tw	tw	NOUN
ap-1255	188	9	,	,	PUNCT
ap-1255	188	10	•	•	ADP
ap-1255	188	11	for	for	ADP
ap-1255	188	12	each	each	DET
ap-1255	188	13	x	x	SYM
ap-1255	188	14	∈	∈	PROPN
ap-1255	188	15	zβ	zβ	NOUN
ap-1255	189	1	we	we	PRON
ap-1255	189	2	have	have	VERB
ap-1255	189	3	φ(x	φ(x	NOUN
ap-1255	189	4	)	)	PUNCT
ap-1255	189	5	∈	∈	PROPN
ap-1255	189	6	inn(tε	inn(tε	NOUN
ap-1255	189	7	)	)	PUNCT
ap-1255	189	8	,	,	PUNCT
ap-1255	189	9	where	where	SCONJ
ap-1255	189	10	ε	ε	PROPN
ap-1255	189	11	is	be	AUX
ap-1255	189	12	the	the	DET
ap-1255	189	13	empty	empty	ADJ
ap-1255	189	14	word	word	NOUN
ap-1255	189	15	and	and	CCONJ
ap-1255	189	16	inn(x	inn(x	PROPN
ap-1255	189	17	)	)	PUNCT
ap-1255	189	18	denotes	denote	VERB
ap-1255	189	19	the	the	DET
ap-1255	189	20	set	set	NOUN
ap-1255	189	21	of	of	ADP
ap-1255	189	22	inner	inner	ADJ
ap-1255	189	23	points	point	NOUN
ap-1255	189	24	of	of	ADP
ap-1255	189	25	x	x	PRON
ap-1255	189	26	;	;	PUNCT
ap-1255	189	27	especially	especially	ADV
ap-1255	189	28	,	,	PUNCT
ap-1255	189	29	the	the	DET
ap-1255	189	30	origin	origin	NOUN
ap-1255	189	31	0	0	NUM
ap-1255	189	32	is	be	AUX
ap-1255	189	33	an	an	DET
ap-1255	189	34	inner	inner	ADJ
ap-1255	189	35	point	point	NOUN
ap-1255	189	36	of	of	ADP
ap-1255	189	37	the	the	DET
ap-1255	189	38	so	so	ADV
ap-1255	189	39	-	-	PUNCT
ap-1255	189	40	called	call	VERB
ap-1255	189	41	central	central	ADJ
ap-1255	189	42	tile	tile	NOUN
ap-1255	189	43	tε	tε	NOUN
ap-1255	189	44	,	,	PUNCT
ap-1255	189	45	•	•	NOUN
ap-1255	189	46	for	for	ADP
ap-1255	189	47	each	each	DET
ap-1255	189	48	tile	tile	NOUN
ap-1255	189	49	tw	tw	NOUN
ap-1255	189	50	we	we	PRON
ap-1255	189	51	have	have	VERB
ap-1255	189	52	inn(tw	inn(tw	ADJ
ap-1255	189	53	)	)	PUNCT
ap-1255	189	54	=	=	SYM
ap-1255	189	55	tw	tw	PROPN
ap-1255	189	56	,	,	PUNCT
ap-1255	189	57	•	•	NOUN
ap-1255	189	58	∂(tw	∂(tw	NOUN
ap-1255	189	59	)	)	PUNCT
ap-1255	189	60	is	be	AUX
ap-1255	189	61	closed	close	VERB
ap-1255	189	62	and	and	CCONJ
ap-1255	189	63	nowhere	nowhere	ADV
ap-1255	189	64	dense	dense	ADJ
ap-1255	189	65	in	in	ADP
ap-1255	189	66	r	r	NOUN
ap-1255	189	67	d−1	d−1	PROPN
ap-1255	189	68	,	,	PUNCT
ap-1255	189	69	where	where	SCONJ
ap-1255	189	70	∂(tw	∂(tw	NOUN
ap-1255	189	71	)	)	PUNCT
ap-1255	189	72	is	be	AUX
ap-1255	189	73	the	the	DET
ap-1255	189	74	set	set	NOUN
ap-1255	189	75	of	of	ADP
ap-1255	189	76	boundary	boundary	ADJ
ap-1255	189	77	elements	element	NOUN
ap-1255	189	78	of	of	ADP
ap-1255	189	79	tw	tw	NOUN
ap-1255	189	80	,	,	PUNCT
ap-1255	189	81	•	•	ADP
ap-1255	189	82	if	if	SCONJ
ap-1255	189	83	dβ(1	dβ(1	PROPN
ap-1255	189	84	)	)	PUNCT
ap-1255	189	85	=	=	NOUN
ap-1255	189	86	t1	t1	NOUN
ap-1255	189	87	·	·	PUNCT
ap-1255	189	88	·	·	PUNCT
ap-1255	189	89	·	·	PUNCT
ap-1255	190	1	tm−11	tm−11	X
ap-1255	190	2	then	then	ADV
ap-1255	190	3	each	each	DET
ap-1255	190	4	tile	tile	NOUN
ap-1255	190	5	tw	tw	NOUN
ap-1255	190	6	is	be	AUX
ap-1255	190	7	arc	arc	NOUN
ap-1255	190	8	-	-	ADJ
ap-1255	190	9	wise	wise	ADJ
ap-1255	190	10	connected	connect	VERB
ap-1255	190	11	.	.	PUNCT
ap-1255	191	1	10	10	NUM
ap-1255	191	2	acta	acta	PROPN
ap-1255	191	3	polytechnica	polytechnica	PROPN
ap-1255	191	4	vol	vol	NOUN
ap-1255	191	5	.	.	PROPN
ap-1255	192	1	50	50	NUM
ap-1255	192	2	no	no	NOUN
ap-1255	192	3	.	.	PUNCT
ap-1255	193	1	5/2010	5/2010	PROPN
ap-1255	193	2	theorem	theorem	NOUN
ap-1255	193	3	8	8	NUM
ap-1255	193	4	(	(	PUNCT
ap-1255	193	5	[	[	X
ap-1255	193	6	2	2	NUM
ap-1255	193	7	]	]	PUNCT
ap-1255	193	8	)	)	PUNCT
ap-1255	193	9	let	let	VERB
ap-1255	193	10	β	β	NOUN
ap-1255	193	11	be	be	AUX
ap-1255	193	12	a	a	DET
ap-1255	193	13	pisot	pisot	ADJ
ap-1255	193	14	unit	unit	NOUN
ap-1255	193	15	of	of	ADP
ap-1255	193	16	degree	degree	NOUN
ap-1255	193	17	d	d	X
ap-1255	193	18	such	such	ADJ
ap-1255	193	19	that	that	SCONJ
ap-1255	193	20	dβ(1	dβ(1	PROPN
ap-1255	193	21	)	)	PUNCT
ap-1255	193	22	=	=	SYM
ap-1255	193	23	t1	t1	NOUN
ap-1255	193	24	.	.	PUNCT
ap-1255	193	25	.	.	PUNCT
ap-1255	193	26	.	.	PUNCT
ap-1255	194	1	tm(tm+1	tm(tm+1	PROPN
ap-1255	194	2	·	·	PUNCT
ap-1255	194	3	·	·	PUNCT
ap-1255	194	4	·	·	PUNCT
ap-1255	195	1	tm+p	tm+p	NUM
ap-1255	195	2	)	)	PUNCT
ap-1255	195	3	ω	ω	PROPN
ap-1255	195	4	with	with	ADP
ap-1255	195	5	m	m	PROPN
ap-1255	195	6	,	,	PUNCT
ap-1255	195	7	p	p	X
ap-1255	195	8	the	the	DET
ap-1255	195	9	smallest	small	ADJ
ap-1255	195	10	possible	possible	ADJ
ap-1255	195	11	.	.	PUNCT
ap-1255	196	1	then	then	ADV
ap-1255	196	2	there	there	PRON
ap-1255	196	3	are	be	VERB
ap-1255	196	4	exactly	exactly	ADV
ap-1255	196	5	m+	m+	NUM
ap-1255	196	6	p	p	NOUN
ap-1255	196	7	different	different	ADJ
ap-1255	196	8	tiles	tile	NOUN
ap-1255	196	9	up	up	ADP
ap-1255	196	10	to	to	ADP
ap-1255	196	11	translation	translation	NOUN
ap-1255	196	12	.	.	PUNCT
ap-1255	197	1	note	note	VERB
ap-1255	197	2	that	that	SCONJ
ap-1255	197	3	q(β	q(β	PROPN
ap-1255	197	4	)	)	PUNCT
ap-1255	197	5	=	=	SYM
ap-1255	197	6	q(−β	q(−β	NOUN
ap-1255	197	7	)	)	PUNCT
ap-1255	197	8	and	and	CCONJ
ap-1255	197	9	z[β	z[β	NOUN
ap-1255	197	10	]	]	X
ap-1255	197	11	=	=	PUNCT
ap-1255	197	12	z[−β	z[−β	PROPN
ap-1255	197	13	]	]	PUNCT
ap-1255	197	14	.	.	PUNCT
ap-1255	198	1	thus	thus	ADV
ap-1255	198	2	the	the	DET
ap-1255	198	3	construction	construction	NOUN
ap-1255	198	4	of	of	ADP
ap-1255	198	5	the	the	DET
ap-1255	198	6	tiling	tiling	NOUN
ap-1255	198	7	associated	associate	VERB
ap-1255	198	8	to	to	ADP
ap-1255	198	9	(	(	PUNCT
ap-1255	198	10	−β)numeration	−β)numeration	NOUN
ap-1255	198	11	follows	follow	VERB
ap-1255	198	12	the	the	DET
ap-1255	198	13	same	same	ADJ
ap-1255	198	14	lines	line	NOUN
ap-1255	198	15	,	,	PUNCT
ap-1255	198	16	the	the	DET
ap-1255	198	17	corresponding	correspond	VERB
ap-1255	198	18	mapping	mapping	NOUN
ap-1255	198	19	φ−	φ−	PROPN
ap-1255	198	20	being	be	AUX
ap-1255	198	21	defined	define	VERB
ap-1255	198	22	using	use	VERB
ap-1255	198	23	isomorphisms	isomorphism	NOUN
ap-1255	198	24	of	of	ADP
ap-1255	198	25	the	the	DET
ap-1255	198	26	extension	extension	NOUN
ap-1255	198	27	fields	field	NOUN
ap-1255	198	28	q(−β	q(−β	NOUN
ap-1255	198	29	)	)	PUNCT
ap-1255	198	30	and	and	CCONJ
ap-1255	198	31	q(−β(j	q(−β(j	NOUN
ap-1255	198	32	)	)	PUNCT
ap-1255	198	33	)	)	PUNCT
ap-1255	198	34	,	,	PUNCT
ap-1255	198	35	and	and	CCONJ
ap-1255	198	36	the	the	DET
ap-1255	198	37	following	follow	VERB
ap-1255	198	38	variant	variant	NOUN
ap-1255	198	39	of	of	ADP
ap-1255	198	40	proposition	proposition	NOUN
ap-1255	198	41	6	6	NUM
ap-1255	198	42	holds	hold	VERB
ap-1255	198	43	;	;	PUNCT
ap-1255	198	44	its	its	PRON
ap-1255	198	45	proof	proof	NOUN
ap-1255	198	46	follows	follow	VERB
ap-1255	198	47	the	the	DET
ap-1255	198	48	same	same	ADJ
ap-1255	198	49	lines	line	NOUN
ap-1255	198	50	as	as	ADP
ap-1255	198	51	in	in	ADP
ap-1255	198	52	the	the	DET
ap-1255	198	53	proof	proof	NOUN
ap-1255	198	54	of	of	ADP
ap-1255	198	55	the	the	DET
ap-1255	198	56	original	original	ADJ
ap-1255	198	57	proposition	proposition	NOUN
ap-1255	198	58	.	.	PUNCT
ap-1255	199	1	proposition	proposition	NOUN
ap-1255	199	2	9	9	NUM
ap-1255	199	3	let	let	VERB
ap-1255	199	4	β	β	PRON
ap-1255	199	5	>	>	X
ap-1255	199	6	1	1	NUM
ap-1255	199	7	be	be	AUX
ap-1255	199	8	a	a	DET
ap-1255	199	9	pisot	pisot	ADJ
ap-1255	199	10	number	number	NOUN
ap-1255	199	11	of	of	ADP
ap-1255	199	12	degree	degree	NOUN
ap-1255	199	13	d.	d.	PROPN
ap-1255	199	14	then	then	ADV
ap-1255	199	15	φ−(z[−β	φ−(z[−β	PROPN
ap-1255	199	16	]	]	PUNCT
ap-1255	199	17	)	)	PUNCT
ap-1255	199	18	is	be	AUX
ap-1255	199	19	dense	dense	ADJ
ap-1255	199	20	in	in	ADP
ap-1255	199	21	the	the	DET
ap-1255	199	22	space	space	NOUN
ap-1255	199	23	r	r	NOUN
ap-1255	199	24	d−1	d−1	PROPN
ap-1255	199	25	.	.	PROPN
ap-1255	199	26	6	6	NUM
ap-1255	199	27	examples	example	NOUN
ap-1255	199	28	of	of	ADP
ap-1255	199	29	tilings	tiling	NOUN
ap-1255	199	30	in	in	ADP
ap-1255	199	31	the	the	DET
ap-1255	199	32	rest	rest	NOUN
ap-1255	199	33	of	of	ADP
ap-1255	199	34	the	the	DET
ap-1255	199	35	paper	paper	NOUN
ap-1255	199	36	we	we	PRON
ap-1255	199	37	provide	provide	VERB
ap-1255	199	38	several	several	ADJ
ap-1255	199	39	examples	example	NOUN
ap-1255	199	40	of	of	ADP
ap-1255	199	41	tilings	tiling	NOUN
ap-1255	199	42	associated	associate	VERB
ap-1255	199	43	with	with	ADP
ap-1255	199	44	β	β	X
ap-1255	199	45	cubic	cubic	ADJ
ap-1255	199	46	pisot	pisot	ADJ
ap-1255	199	47	units	unit	NOUN
ap-1255	199	48	,	,	PUNCT
ap-1255	199	49	i.e.	i.e.	X
ap-1255	199	50	,	,	PUNCT
ap-1255	199	51	the	the	DET
ap-1255	199	52	minimal	minimal	ADJ
ap-1255	199	53	polynomial	polynomial	NOUN
ap-1255	199	54	of	of	ADP
ap-1255	199	55	β	β	PROPN
ap-1255	199	56	is	be	AUX
ap-1255	199	57	of	of	ADP
ap-1255	199	58	the	the	DET
ap-1255	199	59	form	form	NOUN
ap-1255	199	60	x3	x3	ADJ
ap-1255	199	61	−	−	NOUN
ap-1255	199	62	ax2	ax2	NOUN
ap-1255	199	63	−	−	PROPN
ap-1255	199	64	bx	bx	PROPN
ap-1255	199	65	±	±	PROPN
ap-1255	199	66	1	1	NUM
ap-1255	199	67	.	.	PUNCT
ap-1255	200	1	every	every	DET
ap-1255	200	2	time	time	NOUN
ap-1255	200	3	all	all	DET
ap-1255	200	4	the	the	DET
ap-1255	200	5	tiles	tile	NOUN
ap-1255	200	6	tw	tw	NOUN
ap-1255	200	7	with	with	ADP
ap-1255	200	8	w	w	PROPN
ap-1255	200	9	of	of	ADP
ap-1255	200	10	length	length	NOUN
ap-1255	200	11	0	0	NUM
ap-1255	200	12	,	,	PUNCT
ap-1255	200	13	1	1	NUM
ap-1255	200	14	,	,	PUNCT
ap-1255	200	15	2	2	NUM
ap-1255	200	16	are	be	AUX
ap-1255	200	17	plotted	plot	VERB
ap-1255	200	18	.	.	PUNCT
ap-1255	201	1	so	so	ADV
ap-1255	201	2	far	far	ADV
ap-1255	201	3	no	no	DET
ap-1255	201	4	properties	property	NOUN
ap-1255	201	5	of	of	ADP
ap-1255	201	6	tilings	tiling	NOUN
ap-1255	201	7	in	in	ADP
ap-1255	201	8	the	the	DET
ap-1255	201	9	negative	negative	ADJ
ap-1255	201	10	case	case	NOUN
ap-1255	201	11	similar	similar	ADJ
ap-1255	201	12	to	to	ADP
ap-1255	201	13	those	those	PRON
ap-1255	201	14	in	in	ADP
ap-1255	201	15	theorem	theorem	ADJ
ap-1255	201	16	7	7	NUM
ap-1255	201	17	and	and	CCONJ
ap-1255	201	18	theorem	theorem	VERB
ap-1255	201	19	8	8	NUM
ap-1255	201	20	have	have	AUX
ap-1255	201	21	been	be	AUX
ap-1255	201	22	proved	prove	VERB
ap-1255	201	23	.	.	PUNCT
ap-1255	202	1	however	however	ADV
ap-1255	202	2	,	,	PUNCT
ap-1255	202	3	the	the	DET
ap-1255	202	4	following	follow	VERB
ap-1255	202	5	examples	example	NOUN
ap-1255	202	6	demonstrate	demonstrate	VERB
ap-1255	202	7	that	that	SCONJ
ap-1255	202	8	it	it	PRON
ap-1255	202	9	is	be	AUX
ap-1255	202	10	reasonable	reasonable	ADJ
ap-1255	202	11	to	to	PART
ap-1255	202	12	anticipate	anticipate	VERB
ap-1255	202	13	that	that	SCONJ
ap-1255	202	14	most	most	ADJ
ap-1255	202	15	of	of	ADP
ap-1255	202	16	the	the	DET
ap-1255	202	17	properties	property	NOUN
ap-1255	202	18	remain	remain	VERB
ap-1255	202	19	valid	valid	ADJ
ap-1255	202	20	.	.	PUNCT
ap-1255	203	1	on	on	ADP
ap-1255	203	2	the	the	DET
ap-1255	203	3	other	other	ADJ
ap-1255	203	4	hand	hand	NOUN
ap-1255	203	5	,	,	PUNCT
ap-1255	203	6	one	one	PRON
ap-1255	203	7	can	can	AUX
ap-1255	203	8	also	also	ADV
ap-1255	203	9	observe	observe	VERB
ap-1255	203	10	that	that	SCONJ
ap-1255	203	11	for	for	ADP
ap-1255	203	12	a	a	DET
ap-1255	203	13	fixed	fixed	ADJ
ap-1255	203	14	β	β	NOUN
ap-1255	203	15	when	when	SCONJ
ap-1255	203	16	we	we	PRON
ap-1255	203	17	change	change	VERB
ap-1255	203	18	the	the	DET
ap-1255	203	19	β	β	NOUN
ap-1255	203	20	-	-	NOUN
ap-1255	203	21	numeration	numeration	NOUN
ap-1255	203	22	into	into	ADP
ap-1255	203	23	the	the	DET
ap-1255	203	24	(	(	PUNCT
ap-1255	203	25	−β)-numeration	−β)-numeration	NOUN
ap-1255	203	26	(	(	PUNCT
ap-1255	203	27	either	either	CCONJ
ap-1255	203	28	ito	ito	PROPN
ap-1255	203	29	-	-	PROPN
ap-1255	203	30	sadahiro	sadahiro	PROPN
ap-1255	203	31	or	or	CCONJ
ap-1255	203	32	balanced	balanced	ADJ
ap-1255	203	33	)	)	PUNCT
ap-1255	203	34	the	the	DET
ap-1255	203	35	shape	shape	NOUN
ap-1255	203	36	and	and	CCONJ
ap-1255	203	37	form	form	NOUN
ap-1255	203	38	of	of	ADP
ap-1255	203	39	the	the	DET
ap-1255	203	40	tiles	tile	NOUN
ap-1255	203	41	can	can	AUX
ap-1255	203	42	be	be	AUX
ap-1255	203	43	either	either	CCONJ
ap-1255	203	44	preserved	preserve	VERB
ap-1255	203	45	or	or	CCONJ
ap-1255	203	46	changed	change	VERB
ap-1255	203	47	slightly	slightly	ADV
ap-1255	203	48	or	or	CCONJ
ap-1255	203	49	completely	completely	ADV
ap-1255	203	50	.	.	PUNCT
ap-1255	204	1	6.1	6.1	NUM
ap-1255	204	2	minimal	minimal	ADJ
ap-1255	204	3	polynomial	polynomial	ADJ
ap-1255	204	4	x3	x3	NOUN
ap-1255	204	5	−	−	PROPN
ap-1255	204	6	x2	x2	INTJ
ap-1255	204	7	−	−	PROPN
ap-1255	204	8	1	1	NUM
ap-1255	204	9	the	the	DET
ap-1255	204	10	tilings	tiling	NOUN
ap-1255	204	11	associated	associate	VERB
ap-1255	204	12	to	to	ADP
ap-1255	204	13	−β	−β	PROPN
ap-1255	204	14	are	be	AUX
ap-1255	204	15	trivial	trivial	ADJ
ap-1255	204	16	in	in	ADP
ap-1255	204	17	this	this	DET
ap-1255	204	18	case	case	NOUN
ap-1255	204	19	.	.	PUNCT
ap-1255	205	1	indeed	indeed	ADV
ap-1255	205	2	,	,	PUNCT
ap-1255	205	3	d−β(lβ	d−β(lβ	NOUN
ap-1255	205	4	)	)	PUNCT
ap-1255	205	5	=	=	SYM
ap-1255	206	1	1001	1001	NUM
ap-1255	206	2	ω	ω	NUM
ap-1255	206	3	and	and	CCONJ
ap-1255	206	4	db,−β	db,−β	NOUN
ap-1255	206	5	(	(	PUNCT
ap-1255	206	6	−1	−1	NOUN
ap-1255	206	7	2	2	NUM
ap-1255	206	8	)	)	PUNCT
ap-1255	206	9	=	=	SYM
ap-1255	206	10	(	(	PUNCT
ap-1255	206	11	10(−1)(−1	10(−1)(−1	NUM
ap-1255	206	12	)	)	PUNCT
ap-1255	206	13	(	(	PUNCT
ap-1255	206	14	−1)(−1)(−1)010(−1)011	−1)(−1)(−1)010(−1)011	PROPN
ap-1255	206	15	)	)	PUNCT
ap-1255	206	16	ω	ω	NOUN
ap-1255	206	17	,	,	PUNCT
ap-1255	206	18	hence	hence	ADV
ap-1255	206	19	z−β	z−β	PROPN
ap-1255	206	20	=	=	SYM
ap-1255	206	21	zb,−β	zb,−β	PROPN
ap-1255	206	22	=	=	SYM
ap-1255	206	23	{	{	PUNCT
ap-1255	206	24	0	0	NUM
ap-1255	206	25	}	}	PUNCT
ap-1255	206	26	(	(	PUNCT
ap-1255	206	27	cf	cf	NOUN
ap-1255	206	28	.	.	PUNCT
ap-1255	207	1	[	[	X
ap-1255	207	2	3	3	NUM
ap-1255	207	3	,	,	PUNCT
ap-1255	207	4	4	4	NUM
ap-1255	207	5	]	]	NUM
ap-1255	207	6	)	)	PUNCT
ap-1255	207	7	.	.	PUNCT
ap-1255	208	1	6.2	6.2	NUM
ap-1255	208	2	minimal	minimal	ADJ
ap-1255	208	3	polynomial	polynomial	ADJ
ap-1255	208	4	x3	x3	NOUN
ap-1255	208	5	−	−	PROPN
ap-1255	208	6	2x2	2x2	NUM
ap-1255	208	7	−	−	NOUN
ap-1255	208	8	2x	2x	NUM
ap-1255	208	9	−	−	PROPN
ap-1255	208	10	1	1	NUM
ap-1255	208	11	this	this	DET
ap-1255	208	12	β	β	NOUN
ap-1255	208	13	is	be	AUX
ap-1255	208	14	an	an	DET
ap-1255	208	15	example	example	NOUN
ap-1255	208	16	of	of	ADP
ap-1255	208	17	a	a	DET
ap-1255	208	18	base	base	NOUN
ap-1255	208	19	for	for	ADP
ap-1255	208	20	which	which	PRON
ap-1255	208	21	the	the	DET
ap-1255	208	22	three	three	NUM
ap-1255	208	23	considered	consider	VERB
ap-1255	208	24	tilings	tiling	NOUN
ap-1255	208	25	almost	almost	ADV
ap-1255	208	26	do	do	AUX
ap-1255	208	27	not	not	PART
ap-1255	208	28	change	change	VERB
ap-1255	208	29	.	.	PUNCT
ap-1255	209	1	we	we	PRON
ap-1255	209	2	have	have	VERB
ap-1255	209	3	dβ(1	dβ(1	NOUN
ap-1255	209	4	)	)	PUNCT
ap-1255	209	5	=	=	NOUN
ap-1255	209	6	211	211	NUM
ap-1255	209	7	,	,	PUNCT
ap-1255	209	8	d−β(lβ	d−β(lβ	NOUN
ap-1255	209	9	)	)	PUNCT
ap-1255	209	10	=	=	SYM
ap-1255	209	11	201ω	201ω	NOUN
ap-1255	209	12	,	,	PUNCT
ap-1255	209	13	db,−β	db,−β	PROPN
ap-1255	209	14	(	(	PUNCT
ap-1255	209	15	−1	−1	NOUN
ap-1255	209	16	2	2	NUM
ap-1255	209	17	)	)	PUNCT
ap-1255	209	18	=	=	SYM
ap-1255	210	1	(	(	PUNCT
ap-1255	210	2	1(−1)1)ω	1(−1)1)ω	NUM
ap-1255	210	3	.	.	PUNCT
ap-1255	211	1	all	all	DET
ap-1255	211	2	three	three	NUM
ap-1255	211	3	sets	set	VERB
ap-1255	211	4	zβ	zβ	NOUN
ap-1255	211	5	,	,	PUNCT
ap-1255	211	6	z−β	z−β	PROPN
ap-1255	211	7	and	and	CCONJ
ap-1255	211	8	zb,−β	zb,−β	PROPN
ap-1255	211	9	have	have	VERB
ap-1255	211	10	the	the	DET
ap-1255	211	11	same	same	ADJ
ap-1255	211	12	set	set	NOUN
ap-1255	211	13	of	of	ADP
ap-1255	211	14	three	three	NUM
ap-1255	211	15	possible	possible	ADJ
ap-1255	211	16	distances	distance	NOUN
ap-1255	211	17	between	between	ADP
ap-1255	211	18	consecutive	consecutive	ADJ
ap-1255	211	19	elements	element	NOUN
ap-1255	211	20	,	,	PUNCT
ap-1255	211	21	namely	namely	ADV
ap-1255	211	22	{	{	PUNCT
ap-1255	211	23	1	1	NUM
ap-1255	211	24	,	,	PUNCT
ap-1255	211	25	β	β	X
ap-1255	211	26	−	−	NOUN
ap-1255	211	27	2	2	NUM
ap-1255	211	28	,	,	PUNCT
ap-1255	211	29	β2	β2	NOUN
ap-1255	211	30	−	−	PROPN
ap-1255	211	31	2β	2β	NOUN
ap-1255	211	32	−	−	NOUN
ap-1255	211	33	2	2	NUM
ap-1255	211	34	}	}	PUNCT
ap-1255	211	35	.	.	PUNCT
ap-1255	212	1	the	the	DET
ap-1255	212	2	codings	coding	NOUN
ap-1255	212	3	of	of	ADP
ap-1255	212	4	the	the	DET
ap-1255	212	5	distances	distance	NOUN
ap-1255	212	6	in	in	ADP
ap-1255	212	7	these	these	DET
ap-1255	212	8	sets	set	NOUN
ap-1255	212	9	are	be	AUX
ap-1255	212	10	generated	generate	VERB
ap-1255	212	11	by	by	ADP
ap-1255	212	12	substitutions	substitution	NOUN
ap-1255	212	13	which	which	PRON
ap-1255	212	14	are	be	AUX
ap-1255	212	15	pairwise	pairwise	NOUN
ap-1255	212	16	conjugated	conjugate	VERB
ap-1255	212	17	.	.	PUNCT
ap-1255	213	1	recall	recall	VERB
ap-1255	213	2	that	that	SCONJ
ap-1255	213	3	substitutions	substitution	NOUN
ap-1255	213	4	ϕ	ϕ	NOUN
ap-1255	213	5	and	and	CCONJ
ap-1255	213	6	ψ	ψ	X
ap-1255	213	7	over	over	ADP
ap-1255	213	8	an	an	DET
ap-1255	213	9	alphabet	alphabet	NOUN
ap-1255	213	10	a	a	PRON
ap-1255	213	11	are	be	AUX
ap-1255	213	12	said	say	VERB
ap-1255	213	13	to	to	PART
ap-1255	213	14	be	be	AUX
ap-1255	213	15	conjugated	conjugate	VERB
ap-1255	213	16	if	if	SCONJ
ap-1255	213	17	there	there	PRON
ap-1255	213	18	exists	exist	VERB
ap-1255	213	19	a	a	DET
ap-1255	213	20	word	word	NOUN
ap-1255	213	21	w	w	PROPN
ap-1255	213	22	∈	∈	PROPN
ap-1255	213	23	a∗	a∗	NOUN
ap-1255	213	24	such	such	ADJ
ap-1255	213	25	that	that	SCONJ
ap-1255	213	26	ϕ(a	ϕ(a	NOUN
ap-1255	213	27	)	)	PUNCT
ap-1255	214	1	=	=	SYM
ap-1255	214	2	wψ(a)w−1	wψ(a)w−1	PROPN
ap-1255	214	3	for	for	ADP
ap-1255	214	4	all	all	DET
ap-1255	214	5	a	a	DET
ap-1255	214	6	∈	∈	NOUN
ap-1255	214	7	a.	a.	NOUN
ap-1255	214	8	the	the	DET
ap-1255	214	9	tilings	tiling	NOUN
ap-1255	214	10	are	be	AUX
ap-1255	214	11	composed	compose	VERB
ap-1255	214	12	of	of	ADP
ap-1255	214	13	the	the	DET
ap-1255	214	14	same	same	ADJ
ap-1255	214	15	tiles	tile	NOUN
ap-1255	214	16	(	(	PUNCT
ap-1255	214	17	up	up	ADP
ap-1255	214	18	to	to	ADP
ap-1255	214	19	rotation	rotation	NOUN
ap-1255	214	20	)	)	PUNCT
ap-1255	214	21	.	.	PUNCT
ap-1255	215	1	see	see	VERB
ap-1255	215	2	figure	figure	NOUN
ap-1255	215	3	1	1	NUM
ap-1255	215	4	.	.	NUM
ap-1255	215	5	6.3	6.3	NUM
ap-1255	215	6	minimal	minimal	ADJ
ap-1255	215	7	polynomial	polynomial	ADJ
ap-1255	215	8	x3	x3	NOUN
ap-1255	215	9	−	−	NOUN
ap-1255	215	10	3x2	3x2	NUM
ap-1255	216	1	+	+	CCONJ
ap-1255	216	2	x	x	SYM
ap-1255	216	3	−	−	NOUN
ap-1255	216	4	1	1	NUM
ap-1255	216	5	in	in	ADP
ap-1255	216	6	this	this	DET
ap-1255	216	7	case	case	NOUN
ap-1255	216	8	dβ(1	dβ(1	PROPN
ap-1255	216	9	)	)	PUNCT
ap-1255	216	10	=	=	NOUN
ap-1255	216	11	2201	2201	NUM
ap-1255	216	12	,	,	PUNCT
ap-1255	216	13	d−β(lβ	d−β(lβ	NOUN
ap-1255	216	14	)	)	PUNCT
ap-1255	216	15	=	=	PUNCT
ap-1255	216	16	(	(	PUNCT
ap-1255	216	17	201)ω	201)ω	NUM
ap-1255	216	18	,	,	PUNCT
ap-1255	216	19	db,−β	db,−β	PROPN
ap-1255	216	20	(	(	PUNCT
ap-1255	216	21	−1	−1	NOUN
ap-1255	216	22	2	2	NUM
ap-1255	216	23	)	)	PUNCT
ap-1255	216	24	=	=	SYM
ap-1255	216	25	(	(	PUNCT
ap-1255	216	26	1(−1)00)ω	1(−1)00)ω	PROPN
ap-1255	216	27	,	,	PUNCT
ap-1255	216	28	and	and	CCONJ
ap-1255	216	29	again	again	ADV
ap-1255	216	30	all	all	DET
ap-1255	216	31	three	three	NUM
ap-1255	216	32	sets	set	NOUN
ap-1255	216	33	of	of	ADP
ap-1255	216	34	integers	integer	NOUN
ap-1255	216	35	have	have	VERB
ap-1255	216	36	the	the	DET
ap-1255	216	37	same	same	ADJ
ap-1255	216	38	possible	possible	ADJ
ap-1255	216	39	distances	distance	NOUN
ap-1255	216	40	between	between	ADP
ap-1255	216	41	consecutive	consecutive	ADJ
ap-1255	216	42	elements	element	NOUN
ap-1255	216	43	,	,	PUNCT
ap-1255	216	44	δi	δi	PROPN
ap-1255	216	45	∈	∈	PROPN
ap-1255	216	46	{	{	PUNCT
ap-1255	216	47	1	1	NUM
ap-1255	216	48	,	,	PUNCT
ap-1255	216	49	β	β	X
ap-1255	216	50	−	−	NOUN
ap-1255	216	51	2	2	NUM
ap-1255	216	52	,	,	PUNCT
ap-1255	216	53	β2	β2	NOUN
ap-1255	216	54	−	−	PROPN
ap-1255	216	55	2β	2β	NOUN
ap-1255	216	56	−	−	PROPN
ap-1255	216	57	2	2	NUM
ap-1255	216	58	,	,	PUNCT
ap-1255	216	59	β2	β2	NOUN
ap-1255	216	60	−	−	PROPN
ap-1255	216	61	3β	3β	NUM
ap-1255	216	62	+	+	CCONJ
ap-1255	216	63	1	1	NUM
ap-1255	216	64	}	}	PUNCT
ap-1255	216	65	.	.	PUNCT
ap-1255	217	1	however	however	ADV
ap-1255	217	2	,	,	PUNCT
ap-1255	217	3	in	in	ADP
ap-1255	217	4	this	this	DET
ap-1255	217	5	case	case	NOUN
ap-1255	217	6	the	the	DET
ap-1255	217	7	associated	associated	ADJ
ap-1255	217	8	substitutions	substitution	NOUN
ap-1255	217	9	are	be	AUX
ap-1255	217	10	not	not	PART
ap-1255	217	11	conjugated	conjugate	VERB
ap-1255	217	12	(	(	PUNCT
ap-1255	217	13	the	the	DET
ap-1255	217	14	condition	condition	NOUN
ap-1255	217	15	is	be	AUX
ap-1255	217	16	not	not	PART
ap-1255	217	17	fulfilled	fulfil	VERB
ap-1255	217	18	on	on	ADP
ap-1255	217	19	exactly	exactly	ADV
ap-1255	217	20	one	one	NUM
ap-1255	217	21	of	of	ADP
ap-1255	217	22	four	four	NUM
ap-1255	217	23	letters	letter	NOUN
ap-1255	217	24	)	)	PUNCT
ap-1255	217	25	and	and	CCONJ
ap-1255	217	26	even	even	ADV
ap-1255	217	27	though	though	SCONJ
ap-1255	217	28	the	the	DET
ap-1255	217	29	tilings	tiling	NOUN
ap-1255	217	30	do	do	AUX
ap-1255	217	31	look	look	VERB
ap-1255	217	32	similar	similar	ADJ
ap-1255	217	33	,	,	PUNCT
ap-1255	217	34	they	they	PRON
ap-1255	217	35	are	be	AUX
ap-1255	217	36	composed	compose	VERB
ap-1255	217	37	of	of	ADP
ap-1255	217	38	different	different	ADJ
ap-1255	217	39	tiles	tile	NOUN
ap-1255	217	40	.	.	PUNCT
ap-1255	218	1	see	see	VERB
ap-1255	218	2	figure	figure	NOUN
ap-1255	218	3	2	2	NUM
ap-1255	218	4	.	.	NUM
ap-1255	218	5	6.4	6.4	NUM
ap-1255	218	6	minimal	minimal	ADJ
ap-1255	218	7	polynomial	polynomial	ADJ
ap-1255	218	8	x3	x3	NOUN
ap-1255	218	9	−	−	PROPN
ap-1255	218	10	2x2	2x2	NUM
ap-1255	218	11	−	−	NOUN
ap-1255	218	12	1	1	NUM
ap-1255	219	1	this	this	DET
ap-1255	219	2	β	β	NOUN
ap-1255	219	3	is	be	AUX
ap-1255	219	4	an	an	DET
ap-1255	219	5	example	example	NOUN
ap-1255	219	6	of	of	ADP
ap-1255	219	7	a	a	DET
ap-1255	219	8	base	base	NOUN
ap-1255	219	9	for	for	ADP
ap-1255	219	10	which	which	PRON
ap-1255	219	11	two	two	NUM
ap-1255	219	12	tilings	tiling	NOUN
ap-1255	219	13	(	(	PUNCT
ap-1255	219	14	and	and	CCONJ
ap-1255	219	15	the	the	DET
ap-1255	219	16	corresponding	corresponding	ADJ
ap-1255	219	17	properties	property	NOUN
ap-1255	219	18	of	of	ADP
ap-1255	219	19	the	the	DET
ap-1255	219	20	sets	set	NOUN
ap-1255	219	21	of	of	ADP
ap-1255	219	22	integers	integer	NOUN
ap-1255	219	23	)	)	PUNCT
ap-1255	219	24	are	be	AUX
ap-1255	219	25	very	very	ADV
ap-1255	219	26	similar	similar	ADJ
ap-1255	219	27	,	,	PUNCT
ap-1255	219	28	but	but	CCONJ
ap-1255	219	29	the	the	DET
ap-1255	219	30	third	third	ADJ
ap-1255	219	31	tiling	tiling	NOUN
ap-1255	219	32	differs	differ	VERB
ap-1255	219	33	substantially	substantially	ADV
ap-1255	219	34	.	.	PUNCT
ap-1255	220	1	we	we	PRON
ap-1255	220	2	have	have	VERB
ap-1255	220	3	dβ(1	dβ(1	NOUN
ap-1255	220	4	)	)	PUNCT
ap-1255	220	5	=	=	SYM
ap-1255	220	6	201	201	NUM
ap-1255	220	7	,	,	PUNCT
ap-1255	220	8	d−β(lβ	d−β(lβ	NOUN
ap-1255	220	9	)	)	PUNCT
ap-1255	220	10	=	=	PUNCT
ap-1255	221	1	(	(	PUNCT
ap-1255	221	2	2101	2101	NUM
ap-1255	221	3	)	)	PUNCT
ap-1255	221	4	ω	ω	PROPN
ap-1255	221	5	,	,	PUNCT
ap-1255	221	6	db,−β	db,−β	PROPN
ap-1255	221	7	(	(	PUNCT
ap-1255	221	8	−1	−1	NOUN
ap-1255	221	9	2	2	NUM
ap-1255	221	10	)	)	PUNCT
ap-1255	221	11	=	=	SYM
ap-1255	221	12	(	(	PUNCT
ap-1255	221	13	101)ω	101)ω	NUM
ap-1255	221	14	.	.	PUNCT
ap-1255	221	15	11	11	NUM
ap-1255	221	16	acta	acta	PROPN
ap-1255	221	17	polytechnica	polytechnica	PROPN
ap-1255	221	18	vol	vol	NOUN
ap-1255	221	19	.	.	PROPN
ap-1255	222	1	50	50	NUM
ap-1255	222	2	no	no	NOUN
ap-1255	222	3	.	.	PUNCT
ap-1255	223	1	5/2010	5/2010	PRON
ap-1255	223	2	rényi	rényi	PROPN
ap-1255	223	3	case	case	NOUN
ap-1255	223	4	ito	ito	PROPN
ap-1255	223	5	-	-	PUNCT
ap-1255	223	6	sadahiro	sadahiro	PROPN
ap-1255	223	7	case	case	NOUN
ap-1255	223	8	balanced	balanced	ADJ
ap-1255	223	9	case	case	NOUN
ap-1255	223	10	fig	fig	NOUN
ap-1255	223	11	.	.	PUNCT
ap-1255	224	1	1	1	NUM
ap-1255	224	2	:	:	PUNCT
ap-1255	224	3	minimal	minimal	ADJ
ap-1255	224	4	polynomial	polynomial	ADJ
ap-1255	224	5	x3	x3	ADJ
ap-1255	224	6	−	−	PROPN
ap-1255	224	7	2x2	2x2	NUM
ap-1255	224	8	−	−	NOUN
ap-1255	225	1	2x	2x	NUM
ap-1255	225	2	−	−	PROPN
ap-1255	225	3	1	1	NUM
ap-1255	225	4	12	12	NUM
ap-1255	225	5	acta	acta	PROPN
ap-1255	225	6	polytechnica	polytechnica	PROPN
ap-1255	225	7	vol	vol	NOUN
ap-1255	225	8	.	.	PROPN
ap-1255	226	1	50	50	NUM
ap-1255	226	2	no	no	NOUN
ap-1255	226	3	.	.	PUNCT
ap-1255	227	1	5/2010	5/2010	PRON
ap-1255	227	2	rényi	rényi	PROPN
ap-1255	227	3	case	case	NOUN
ap-1255	227	4	ito	ito	PROPN
ap-1255	227	5	-	-	PUNCT
ap-1255	227	6	sadahiro	sadahiro	PROPN
ap-1255	227	7	case	case	NOUN
ap-1255	227	8	balanced	balanced	ADJ
ap-1255	227	9	case	case	NOUN
ap-1255	227	10	fig	fig	NOUN
ap-1255	227	11	.	.	PUNCT
ap-1255	228	1	2	2	NUM
ap-1255	228	2	:	:	PUNCT
ap-1255	228	3	minimal	minimal	ADJ
ap-1255	228	4	polynomial	polynomial	ADJ
ap-1255	228	5	x3	x3	ADJ
ap-1255	228	6	−	−	NOUN
ap-1255	228	7	3x2	3x2	NUM
ap-1255	229	1	+	+	CCONJ
ap-1255	229	2	x	x	SYM
ap-1255	229	3	−	−	NOUN
ap-1255	229	4	1	1	NUM
ap-1255	229	5	13	13	NUM
ap-1255	229	6	acta	acta	PROPN
ap-1255	229	7	polytechnica	polytechnica	PROPN
ap-1255	229	8	vol	vol	NOUN
ap-1255	229	9	.	.	PROPN
ap-1255	230	1	50	50	NUM
ap-1255	230	2	no	no	NOUN
ap-1255	230	3	.	.	PUNCT
ap-1255	231	1	5/2010	5/2010	PRON
ap-1255	231	2	rényi	rényi	PROPN
ap-1255	231	3	case	case	NOUN
ap-1255	231	4	ito	ito	PROPN
ap-1255	231	5	-	-	PUNCT
ap-1255	231	6	sadahiro	sadahiro	PROPN
ap-1255	231	7	case	case	NOUN
ap-1255	231	8	balanced	balanced	ADJ
ap-1255	231	9	case	case	NOUN
ap-1255	231	10	fig	fig	NOUN
ap-1255	231	11	.	.	PUNCT
ap-1255	232	1	3	3	NUM
ap-1255	232	2	:	:	PUNCT
ap-1255	232	3	minimal	minimal	ADJ
ap-1255	232	4	polynomial	polynomial	ADJ
ap-1255	232	5	x3	x3	ADJ
ap-1255	232	6	−	−	PROPN
ap-1255	232	7	2x2	2x2	NUM
ap-1255	232	8	−	−	NOUN
ap-1255	232	9	1	1	NUM
ap-1255	232	10	14	14	NUM
ap-1255	232	11	acta	acta	PROPN
ap-1255	232	12	polytechnica	polytechnica	PROPN
ap-1255	232	13	vol	vol	NOUN
ap-1255	232	14	.	.	PROPN
ap-1255	233	1	50	50	NUM
ap-1255	233	2	no	no	NOUN
ap-1255	233	3	.	.	PUNCT
ap-1255	234	1	5/2010	5/2010	PRON
ap-1255	234	2	rényi	rényi	PROPN
ap-1255	234	3	case	case	NOUN
ap-1255	234	4	ito	ito	PROPN
ap-1255	234	5	-	-	PUNCT
ap-1255	234	6	sadahiro	sadahiro	PROPN
ap-1255	234	7	case	case	NOUN
ap-1255	234	8	balanced	balanced	ADJ
ap-1255	234	9	case	case	NOUN
ap-1255	234	10	fig	fig	NOUN
ap-1255	234	11	.	.	PUNCT
ap-1255	235	1	4	4	NUM
ap-1255	235	2	:	:	PUNCT
ap-1255	235	3	minimal	minimal	ADJ
ap-1255	235	4	polynomial	polynomial	ADJ
ap-1255	235	5	x3	x3	ADJ
ap-1255	235	6	−	−	PROPN
ap-1255	235	7	3x2	3x2	NUM
ap-1255	236	1	+	+	CCONJ
ap-1255	236	2	2x	2x	NUM
ap-1255	236	3	−	−	PROPN
ap-1255	236	4	1	1	NUM
ap-1255	236	5	the	the	DET
ap-1255	236	6	sets	set	NOUN
ap-1255	236	7	zβ	zβ	PROPN
ap-1255	236	8	and	and	CCONJ
ap-1255	236	9	zb,−β	zb,−β	PROPN
ap-1255	236	10	have	have	VERB
ap-1255	236	11	the	the	DET
ap-1255	236	12	same	same	ADJ
ap-1255	236	13	set	set	NOUN
ap-1255	236	14	of	of	ADP
ap-1255	236	15	distances	distance	NOUN
ap-1255	236	16	{	{	PUNCT
ap-1255	236	17	1	1	NUM
ap-1255	236	18	,	,	PUNCT
ap-1255	236	19	β	β	X
ap-1255	236	20	−	−	NOUN
ap-1255	236	21	2	2	NUM
ap-1255	236	22	,	,	PUNCT
ap-1255	236	23	β2−	β2−	ADJ
ap-1255	236	24	2β	2β	NOUN
ap-1255	236	25	}	}	PUNCT
ap-1255	236	26	,	,	PUNCT
ap-1255	236	27	however	however	ADV
ap-1255	236	28	the	the	DET
ap-1255	236	29	associated	associated	ADJ
ap-1255	236	30	substitutions	substitution	NOUN
ap-1255	236	31	are	be	AUX
ap-1255	236	32	not	not	PART
ap-1255	236	33	conjugated	conjugate	VERB
ap-1255	236	34	.	.	PUNCT
ap-1255	237	1	on	on	ADP
ap-1255	237	2	the	the	DET
ap-1255	237	3	other	other	ADJ
ap-1255	237	4	hand	hand	NOUN
ap-1255	237	5	there	there	PRON
ap-1255	237	6	are	be	VERB
ap-1255	237	7	five	five	NUM
ap-1255	237	8	distances	distance	NOUN
ap-1255	237	9	between	between	ADP
ap-1255	237	10	consecutive	consecutive	ADJ
ap-1255	237	11	elements	element	NOUN
ap-1255	237	12	in	in	ADP
ap-1255	237	13	the	the	DET
ap-1255	237	14	set	set	NOUN
ap-1255	237	15	z−β	z−β	PROPN
ap-1255	237	16	,	,	PUNCT
ap-1255	237	17	namely	namely	ADV
ap-1255	237	18	{	{	PUNCT
ap-1255	237	19	1	1	NUM
ap-1255	237	20	,	,	PUNCT
ap-1255	237	21	β2	β2	NOUN
ap-1255	237	22	−	−	PROPN
ap-1255	237	23	β	β	NOUN
ap-1255	237	24	−	−	PROPN
ap-1255	237	25	1	1	NUM
ap-1255	237	26	,	,	PUNCT
ap-1255	237	27	β	β	X
ap-1255	237	28	−	−	NOUN
ap-1255	237	29	1	1	NUM
ap-1255	237	30	,	,	PUNCT
ap-1255	237	31	β	β	X
ap-1255	237	32	,	,	PUNCT
ap-1255	237	33	β2	β2	NOUN
ap-1255	237	34	−	−	NOUN
ap-1255	237	35	β	β	NOUN
ap-1255	237	36	}	}	PUNCT
ap-1255	237	37	.	.	PUNCT
ap-1255	238	1	the	the	DET
ap-1255	238	2	forms	form	NOUN
ap-1255	238	3	of	of	ADP
ap-1255	238	4	the	the	DET
ap-1255	238	5	tilings	tiling	NOUN
ap-1255	238	6	comply	comply	ADV
ap-1255	238	7	:	:	PUNCT
ap-1255	238	8	the	the	DET
ap-1255	238	9	tiling	tiling	NOUN
ap-1255	238	10	in	in	ADP
ap-1255	238	11	the	the	DET
ap-1255	238	12	rényi	rényi	PROPN
ap-1255	238	13	case	case	NOUN
ap-1255	238	14	and	and	CCONJ
ap-1255	238	15	the	the	DET
ap-1255	238	16	tiling	tiling	NOUN
ap-1255	238	17	in	in	ADP
ap-1255	238	18	the	the	DET
ap-1255	238	19	balanced	balanced	ADJ
ap-1255	238	20	case	case	NOUN
ap-1255	238	21	are	be	AUX
ap-1255	238	22	somewhat	somewhat	ADV
ap-1255	238	23	similar	similar	ADJ
ap-1255	238	24	,	,	PUNCT
ap-1255	238	25	but	but	CCONJ
ap-1255	238	26	the	the	DET
ap-1255	238	27	tiling	tiling	NOUN
ap-1255	238	28	in	in	ADP
ap-1255	238	29	the	the	DET
ap-1255	238	30	ito	ito	PROPN
ap-1255	238	31	-	-	PROPN
ap-1255	238	32	sadahiro	sadahiro	NOUN
ap-1255	238	33	case	case	NOUN
ap-1255	238	34	is	be	AUX
ap-1255	238	35	completely	completely	ADV
ap-1255	238	36	different	different	ADJ
ap-1255	238	37	.	.	PUNCT
ap-1255	239	1	see	see	VERB
ap-1255	239	2	figure	figure	NOUN
ap-1255	239	3	3	3	NUM
ap-1255	239	4	.	.	X
ap-1255	239	5	6.5	6.5	NUM
ap-1255	239	6	minimal	minimal	ADJ
ap-1255	239	7	polynomial	polynomial	ADJ
ap-1255	239	8	x3	x3	NOUN
ap-1255	239	9	−	−	PROPN
ap-1255	239	10	3x2	3x2	NUM
ap-1255	240	1	+	+	CCONJ
ap-1255	240	2	2x	2x	NUM
ap-1255	240	3	−	−	PROPN
ap-1255	240	4	1	1	NUM
ap-1255	240	5	the	the	DET
ap-1255	240	6	last	last	ADJ
ap-1255	240	7	example	example	NOUN
ap-1255	240	8	demonstrates	demonstrate	VERB
ap-1255	240	9	that	that	SCONJ
ap-1255	240	10	the	the	DET
ap-1255	240	11	tiling	tiling	NOUN
ap-1255	240	12	can	can	AUX
ap-1255	240	13	change	change	VERB
ap-1255	240	14	fundamentally	fundamentally	ADV
ap-1255	240	15	when	when	SCONJ
ap-1255	240	16	considering	consider	VERB
ap-1255	240	17	different	different	ADJ
ap-1255	240	18	numeration	numeration	NOUN
ap-1255	240	19	systems	system	NOUN
ap-1255	240	20	with	with	ADP
ap-1255	240	21	fixed	fixed	ADJ
ap-1255	240	22	β	β	NOUN
ap-1255	240	23	.	.	PUNCT
ap-1255	241	1	in	in	ADP
ap-1255	241	2	this	this	DET
ap-1255	241	3	case	case	NOUN
ap-1255	241	4	dβ(1	dβ(1	PROPN
ap-1255	241	5	)	)	PUNCT
ap-1255	241	6	=	=	SYM
ap-1255	241	7	201ω	201ω	PROPN
ap-1255	241	8	,	,	PUNCT
ap-1255	241	9	d−β(lβ	d−β(lβ	NOUN
ap-1255	241	10	)	)	PUNCT
ap-1255	241	11	=	=	PUNCT
ap-1255	241	12	(	(	PUNCT
ap-1255	241	13	211)ω	211)ω	NUM
ap-1255	241	14	,	,	PUNCT
ap-1255	241	15	db,−β	db,−β	PROPN
ap-1255	241	16	(	(	PUNCT
ap-1255	241	17	−1	−1	NOUN
ap-1255	241	18	2	2	NUM
ap-1255	241	19	)	)	PUNCT
ap-1255	241	20	=	=	SYM
ap-1255	241	21	(	(	PUNCT
ap-1255	241	22	1010(−1)(−1)(−1)(−1)0(−1)0111)ω	1010(−1)(−1)(−1)(−1)0(−1)0111)ω	NUM
ap-1255	241	23	,	,	PUNCT
ap-1255	241	24	there	there	PRON
ap-1255	241	25	are	be	VERB
ap-1255	241	26	three	three	NUM
ap-1255	241	27	distances	distance	NOUN
ap-1255	241	28	between	between	ADP
ap-1255	241	29	consecutive	consecutive	ADJ
ap-1255	241	30	elements	element	NOUN
ap-1255	241	31	in	in	ADP
ap-1255	241	32	the	the	DET
ap-1255	241	33	set	set	NOUN
ap-1255	241	34	zβ	zβ	PROPN
ap-1255	241	35	,	,	PUNCT
ap-1255	241	36	four	four	NUM
ap-1255	241	37	in	in	ADP
ap-1255	241	38	the	the	DET
ap-1255	241	39	set	set	NOUN
ap-1255	241	40	z−β	z−β	PROPN
ap-1255	241	41	and	and	CCONJ
ap-1255	241	42	seven	seven	NUM
ap-1255	241	43	in	in	ADP
ap-1255	241	44	the	the	DET
ap-1255	241	45	set	set	NOUN
ap-1255	241	46	zb,−β	zb,−β	PROPN
ap-1255	241	47	.	.	PUNCT
ap-1255	242	1	the	the	DET
ap-1255	242	2	tilings	tiling	NOUN
ap-1255	242	3	are	be	AUX
ap-1255	242	4	completely	completely	ADV
ap-1255	242	5	different	different	ADJ
ap-1255	242	6	.	.	PUNCT
ap-1255	243	1	see	see	VERB
ap-1255	243	2	figure	figure	NOUN
ap-1255	243	3	4	4	NUM
ap-1255	243	4	.	.	NOUN
ap-1255	243	5	15	15	NUM
ap-1255	243	6	acta	acta	PROPN
ap-1255	243	7	polytechnica	polytechnica	PROPN
ap-1255	243	8	vol	vol	NOUN
ap-1255	243	9	.	.	PROPN
ap-1255	244	1	50	50	NUM
ap-1255	244	2	no	no	NOUN
ap-1255	244	3	.	.	PUNCT
ap-1255	245	1	5/2010	5/2010	NUM
ap-1255	245	2	7	7	NUM
ap-1255	245	3	conclusion	conclusion	NOUN
ap-1255	245	4	due	due	ADP
ap-1255	245	5	to	to	ADP
ap-1255	245	6	the	the	DET
ap-1255	245	7	similar	similar	ADJ
ap-1255	245	8	nature	nature	NOUN
ap-1255	245	9	of	of	ADP
ap-1255	245	10	β	β	NOUN
ap-1255	245	11	-	-	NOUN
ap-1255	245	12	numeration	numeration	NOUN
ap-1255	245	13	and	and	CCONJ
ap-1255	245	14	(	(	PUNCT
ap-1255	245	15	−β)-numeration	−β)-numeration	NOUN
ap-1255	245	16	,	,	PUNCT
ap-1255	245	17	the	the	DET
ap-1255	245	18	transfer	transfer	NOUN
ap-1255	245	19	of	of	ADP
ap-1255	245	20	the	the	DET
ap-1255	245	21	construction	construction	NOUN
ap-1255	245	22	of	of	ADP
ap-1255	245	23	the	the	DET
ap-1255	245	24	tiling	tiling	NOUN
ap-1255	245	25	of	of	ADP
ap-1255	245	26	a	a	DET
ap-1255	245	27	space	space	NOUN
ap-1255	245	28	due	due	ADP
ap-1255	245	29	to	to	ADP
ap-1255	245	30	thurston	thurston	PROPN
ap-1255	245	31	into	into	ADP
ap-1255	245	32	the	the	DET
ap-1255	245	33	framework	framework	NOUN
ap-1255	245	34	of	of	ADP
ap-1255	245	35	(	(	PUNCT
ap-1255	245	36	−β)-numeration	−β)-numeration	NOUN
ap-1255	245	37	is	be	AUX
ap-1255	245	38	quite	quite	ADV
ap-1255	245	39	straightforward	straightforward	ADJ
ap-1255	245	40	.	.	PUNCT
ap-1255	246	1	in	in	ADP
ap-1255	246	2	this	this	DET
ap-1255	246	3	paper	paper	NOUN
ap-1255	246	4	we	we	PRON
ap-1255	246	5	have	have	AUX
ap-1255	246	6	provided	provide	VERB
ap-1255	246	7	several	several	ADJ
ap-1255	246	8	examples	example	NOUN
ap-1255	246	9	of	of	ADP
ap-1255	246	10	these	these	DET
ap-1255	246	11	tilings	tiling	NOUN
ap-1255	246	12	(	(	PUNCT
ap-1255	246	13	for	for	ADP
ap-1255	246	14	both	both	CCONJ
ap-1255	246	15	the	the	DET
ap-1255	246	16	ito	ito	PROPN
ap-1255	246	17	-	-	PROPN
ap-1255	246	18	sadahiro	sadahiro	PROPN
ap-1255	246	19	definition	definition	NOUN
ap-1255	246	20	and	and	CCONJ
ap-1255	246	21	the	the	DET
ap-1255	246	22	balanced	balanced	ADJ
ap-1255	246	23	definition	definition	NOUN
ap-1255	246	24	of	of	ADP
ap-1255	246	25	the	the	DET
ap-1255	246	26	−(β)-transformation	−(β)-transformation	NOUN
ap-1255	246	27	)	)	PUNCT
ap-1255	246	28	.	.	PUNCT
ap-1255	247	1	although	although	SCONJ
ap-1255	247	2	the	the	DET
ap-1255	247	3	shape	shape	NOUN
ap-1255	247	4	and	and	CCONJ
ap-1255	247	5	form	form	NOUN
ap-1255	247	6	of	of	ADP
ap-1255	247	7	tiling	tile	VERB
ap-1255	247	8	can	can	AUX
ap-1255	247	9	change	change	VERB
ap-1255	247	10	dramatically	dramatically	ADV
ap-1255	247	11	when	when	SCONJ
ap-1255	247	12	one	one	NUM
ap-1255	247	13	changes	change	NOUN
ap-1255	247	14	(	(	PUNCT
ap-1255	247	15	for	for	ADP
ap-1255	247	16	a	a	DET
ap-1255	247	17	fixed	fix	VERB
ap-1255	247	18	β	β	NOUN
ap-1255	247	19	)	)	PUNCT
ap-1255	247	20	the	the	DET
ap-1255	247	21	β	β	NOUN
ap-1255	247	22	-	-	NOUN
ap-1255	247	23	numeration	numeration	NOUN
ap-1255	247	24	into	into	ADP
ap-1255	247	25	the	the	DET
ap-1255	247	26	−(β)-numeration	−(β)-numeration	NOUN
ap-1255	247	27	,	,	PUNCT
ap-1255	247	28	in	in	ADP
ap-1255	247	29	general	general	ADJ
ap-1255	247	30	the	the	DET
ap-1255	247	31	examples	example	NOUN
ap-1255	247	32	demonstrate	demonstrate	VERB
ap-1255	247	33	that	that	SCONJ
ap-1255	247	34	the	the	DET
ap-1255	247	35	validity	validity	NOUN
ap-1255	247	36	of	of	ADP
ap-1255	247	37	most	most	ADJ
ap-1255	247	38	of	of	ADP
ap-1255	247	39	the	the	DET
ap-1255	247	40	properties	property	NOUN
ap-1255	247	41	derived	derive	VERB
ap-1255	247	42	by	by	ADP
ap-1255	247	43	akiyama	akiyama	PROPN
ap-1255	247	44	and	and	CCONJ
ap-1255	247	45	praggastis	praggastis	PROPN
ap-1255	247	46	in	in	ADP
ap-1255	247	47	the	the	DET
ap-1255	247	48	positive	positive	ADJ
ap-1255	247	49	case	case	NOUN
ap-1255	247	50	should	should	AUX
ap-1255	247	51	be	be	AUX
ap-1255	247	52	preserved	preserve	VERB
ap-1255	247	53	.	.	PUNCT
ap-1255	248	1	it	it	PRON
ap-1255	248	2	remains	remain	VERB
ap-1255	248	3	an	an	DET
ap-1255	248	4	open	open	ADJ
ap-1255	248	5	question	question	NOUN
ap-1255	248	6	to	to	PART
ap-1255	248	7	provide	provide	VERB
ap-1255	248	8	proofs	proof	NOUN
ap-1255	248	9	of	of	ADP
ap-1255	248	10	such	such	ADJ
ap-1255	248	11	properties	property	NOUN
ap-1255	248	12	.	.	PUNCT
ap-1255	249	1	acknowledgement	acknowledgement	NOUN
ap-1255	249	2	we	we	PRON
ap-1255	249	3	acknowledge	acknowledge	VERB
ap-1255	249	4	financial	financial	ADJ
ap-1255	249	5	support	support	NOUN
ap-1255	249	6	from	from	ADP
ap-1255	249	7	czech	czech	PROPN
ap-1255	249	8	science	science	NOUN
ap-1255	249	9	foundation	foundation	NOUN
ap-1255	249	10	grant	grant	VERB
ap-1255	249	11	201/09/0584	201/09/0584	NUM
ap-1255	249	12	and	and	CCONJ
ap-1255	249	13	from	from	ADP
ap-1255	249	14	grants	grant	NOUN
ap-1255	249	15	msm	msm	NOUN
ap-1255	249	16	6840770039	6840770039	NUM
ap-1255	249	17	and	and	CCONJ
ap-1255	249	18	lc06002	lc06002	NOUN
ap-1255	249	19	of	of	ADP
ap-1255	249	20	the	the	DET
ap-1255	249	21	ministry	ministry	PROPN
ap-1255	249	22	of	of	ADP
ap-1255	249	23	education	education	PROPN
ap-1255	249	24	,	,	PUNCT
ap-1255	249	25	youth	youth	NOUN
ap-1255	249	26	,	,	PUNCT
ap-1255	249	27	and	and	CCONJ
ap-1255	249	28	sports	sport	NOUN
ap-1255	249	29	of	of	ADP
ap-1255	249	30	the	the	DET
ap-1255	249	31	czech	czech	PROPN
ap-1255	249	32	republic	republic	NOUN
ap-1255	249	33	.	.	PUNCT
ap-1255	250	1	references	reference	NOUN
ap-1255	250	2	[	[	X
ap-1255	250	3	1	1	NUM
ap-1255	250	4	]	]	PUNCT
ap-1255	250	5	akiyama	akiyama	NOUN
ap-1255	250	6	,	,	PUNCT
ap-1255	250	7	s	s	PART
ap-1255	250	8	:	:	PUNCT
ap-1255	250	9	self	self	NOUN
ap-1255	250	10	affine	affine	NOUN
ap-1255	250	11	tiling	tile	VERB
ap-1255	250	12	and	and	CCONJ
ap-1255	250	13	pisot	pisot	ADJ
ap-1255	250	14	numeration	numeration	NOUN
ap-1255	250	15	system	system	NOUN
ap-1255	250	16	.	.	PUNCT
ap-1255	251	1	in	in	ADP
ap-1255	251	2	number	number	NOUN
ap-1255	251	3	theory	theory	NOUN
ap-1255	251	4	and	and	CCONJ
ap-1255	251	5	its	its	PRON
ap-1255	251	6	applications	application	NOUN
ap-1255	251	7	(	(	PUNCT
ap-1255	251	8	kyoto	kyoto	NOUN
ap-1255	251	9	,	,	PUNCT
ap-1255	251	10	1997	1997	NUM
ap-1255	251	11	)	)	PUNCT
ap-1255	251	12	,	,	PUNCT
ap-1255	251	13	k.	k.	PROPN
ap-1255	251	14	győry	győry	PROPN
ap-1255	251	15	and	and	CCONJ
ap-1255	251	16	s.	s.	PROPN
ap-1255	251	17	kanemitsu	kanemitsu	PROPN
ap-1255	251	18	,	,	PUNCT
ap-1255	251	19	(	(	PUNCT
ap-1255	251	20	eds	ed	NOUN
ap-1255	251	21	.	.	PUNCT
ap-1255	251	22	)	)	PUNCT
ap-1255	251	23	,	,	PUNCT
ap-1255	251	24	vol	vol	NOUN
ap-1255	251	25	.	.	PROPN
ap-1255	251	26	2	2	NUM
ap-1255	251	27	of	of	ADP
ap-1255	251	28	dev	dev	PROPN
ap-1255	251	29	.	.	PUNCT
ap-1255	251	30	math	math	PROPN
ap-1255	251	31	.	.	PUNCT
ap-1255	251	32	,	,	PUNCT
ap-1255	251	33	kluwer	kluwer	PROPN
ap-1255	251	34	acad	acad	PROPN
ap-1255	251	35	.	.	PUNCT
ap-1255	252	1	publ	publ	PROPN
ap-1255	252	2	.	.	PUNCT
ap-1255	253	1	1999	1999	NUM
ap-1255	253	2	,	,	PUNCT
ap-1255	253	3	7–17	7–17	PROPN
ap-1255	253	4	.	.	PUNCT
ap-1255	254	1	[	[	X
ap-1255	254	2	2	2	NUM
ap-1255	254	3	]	]	PUNCT
ap-1255	254	4	akiyama	akiyama	NOUN
ap-1255	254	5	,	,	PUNCT
ap-1255	254	6	s.	s.	PROPN
ap-1255	254	7	:	:	PUNCT
ap-1255	254	8	on	on	ADP
ap-1255	254	9	the	the	DET
ap-1255	254	10	boundary	boundary	NOUN
ap-1255	254	11	of	of	ADP
ap-1255	254	12	self	self	NOUN
ap-1255	254	13	affine	affine	NOUN
ap-1255	254	14	tilings	tiling	NOUN
ap-1255	254	15	generated	generate	VERB
ap-1255	254	16	by	by	ADP
ap-1255	254	17	pisot	pisot	ADJ
ap-1255	254	18	numbers	number	NOUN
ap-1255	254	19	.	.	PUNCT
ap-1255	255	1	j.	j.	PROPN
ap-1255	255	2	math	math	PROPN
ap-1255	255	3	.	.	PUNCT
ap-1255	256	1	soc	soc	PROPN
ap-1255	256	2	.	.	PUNCT
ap-1255	257	1	japan	japan	PROPN
ap-1255	257	2	54	54	NUM
ap-1255	257	3	,	,	PUNCT
ap-1255	257	4	2002	2002	NUM
ap-1255	257	5	,	,	PUNCT
ap-1255	257	6	283–308	283–308	NUM
ap-1255	257	7	.	.	PUNCT
ap-1255	258	1	[	[	X
ap-1255	258	2	3	3	NUM
ap-1255	258	3	]	]	PUNCT
ap-1255	258	4	ambrož	ambrož	NOUN
ap-1255	258	5	,	,	PUNCT
ap-1255	258	6	p.	p.	NOUN
ap-1255	258	7	,	,	PUNCT
ap-1255	258	8	dombek	dombek	PROPN
ap-1255	258	9	,	,	PUNCT
ap-1255	258	10	d.	d.	PROPN
ap-1255	258	11	,	,	PUNCT
ap-1255	258	12	masáková	masáková	PROPN
ap-1255	258	13	,	,	PUNCT
ap-1255	258	14	z.	z.	PROPN
ap-1255	258	15	,	,	PUNCT
ap-1255	258	16	pelantová	pelantová	PROPN
ap-1255	258	17	,	,	PUNCT
ap-1255	258	18	e.	e.	PROPN
ap-1255	258	19	:	:	PUNCT
ap-1255	258	20	numbers	number	NOUN
ap-1255	258	21	with	with	ADP
ap-1255	258	22	integer	integer	NOUN
ap-1255	258	23	expansions	expansion	NOUN
ap-1255	258	24	in	in	ADP
ap-1255	258	25	the	the	DET
ap-1255	258	26	numeration	numeration	NOUN
ap-1255	258	27	system	system	NOUN
ap-1255	258	28	with	with	ADP
ap-1255	258	29	negative	negative	ADJ
ap-1255	258	30	base	base	NOUN
ap-1255	258	31	.	.	PUNCT
ap-1255	259	1	submitted	submit	VERB
ap-1255	259	2	to	to	ADP
ap-1255	259	3	acta	acta	PROPN
ap-1255	259	4	arithmetica	arithmetica	PROPN
ap-1255	259	5	.	.	PUNCT
ap-1255	260	1	[	[	X
ap-1255	260	2	4	4	NUM
ap-1255	260	3	]	]	X
ap-1255	260	4	dombek	dombek	PROPN
ap-1255	260	5	,	,	PUNCT
ap-1255	260	6	d.	d.	PROPN
ap-1255	260	7	:	:	PUNCT
ap-1255	260	8	beta	beta	NOUN
ap-1255	260	9	-	-	PUNCT
ap-1255	260	10	numeration	numeration	NOUN
ap-1255	260	11	systems	system	NOUN
ap-1255	260	12	with	with	ADP
ap-1255	260	13	negative	negative	ADJ
ap-1255	260	14	base	base	NOUN
ap-1255	260	15	.	.	PUNCT
ap-1255	261	1	master	master	NOUN
ap-1255	261	2	’s	’s	PART
ap-1255	261	3	thesis	thesis	NOUN
ap-1255	261	4	,	,	PUNCT
ap-1255	261	5	czech	czech	PROPN
ap-1255	261	6	technical	technical	PROPN
ap-1255	261	7	university	university	PROPN
ap-1255	261	8	in	in	ADP
ap-1255	261	9	prague	prague	PROPN
ap-1255	261	10	,	,	PUNCT
ap-1255	261	11	2010	2010	NUM
ap-1255	261	12	.	.	PUNCT
ap-1255	262	1	[	[	X
ap-1255	262	2	5	5	NUM
ap-1255	262	3	]	]	X
ap-1255	262	4	fabre	fabre	PROPN
ap-1255	262	5	,	,	PUNCT
ap-1255	262	6	s.	s.	PROPN
ap-1255	262	7	:	:	PUNCT
ap-1255	262	8	substitutions	substitution	NOUN
ap-1255	262	9	et	et	NOUN
ap-1255	262	10	β	β	X
ap-1255	262	11	-	-	PUNCT
ap-1255	262	12	systèmes	systèmes	PRON
ap-1255	262	13	de	de	X
ap-1255	262	14	numération	numération	PROPN
ap-1255	262	15	.	.	PUNCT
ap-1255	263	1	theoret	theoret	ADJ
ap-1255	263	2	.	.	PUNCT
ap-1255	264	1	comput	comput	NOUN
ap-1255	264	2	.	.	PUNCT
ap-1255	265	1	sci	sci	PROPN
ap-1255	265	2	.	.	PROPN
ap-1255	265	3	137	137	NUM
ap-1255	265	4	,	,	PUNCT
ap-1255	265	5	1995	1995	NUM
ap-1255	265	6	,	,	PUNCT
ap-1255	265	7	219–236	219–236	NUM
ap-1255	265	8	.	.	PUNCT
ap-1255	266	1	[	[	X
ap-1255	266	2	6	6	NUM
ap-1255	266	3	]	]	SYM
ap-1255	266	4	ito	ito	PROPN
ap-1255	266	5	,	,	PUNCT
ap-1255	266	6	s.	s.	PROPN
ap-1255	266	7	,	,	PUNCT
ap-1255	266	8	sadahiro	sadahiro	PROPN
ap-1255	266	9	,	,	PUNCT
ap-1255	266	10	t.	t.	PROPN
ap-1255	266	11	:	:	PUNCT
ap-1255	266	12	beta	beta	NOUN
ap-1255	266	13	-	-	PUNCT
ap-1255	266	14	expansions	expansion	NOUN
ap-1255	266	15	with	with	ADP
ap-1255	266	16	negative	negative	ADJ
ap-1255	266	17	bases	basis	NOUN
ap-1255	266	18	.	.	PUNCT
ap-1255	267	1	integers	integer	NOUN
ap-1255	267	2	,	,	PUNCT
ap-1255	267	3	9	9	NUM
ap-1255	267	4	,	,	PUNCT
ap-1255	267	5	2009	2009	NUM
ap-1255	267	6	,	,	PUNCT
ap-1255	267	7	a22	a22	PROPN
ap-1255	267	8	,	,	PUNCT
ap-1255	267	9	239–259	239–259	NUM
ap-1255	267	10	.	.	PUNCT
ap-1255	268	1	[	[	X
ap-1255	268	2	7	7	NUM
ap-1255	268	3	]	]	X
ap-1255	268	4	parry	parry	PROPN
ap-1255	268	5	,	,	PUNCT
ap-1255	268	6	w.	w.	PROPN
ap-1255	268	7	:	:	PUNCT
ap-1255	268	8	on	on	ADP
ap-1255	268	9	the	the	DET
ap-1255	268	10	β	β	NOUN
ap-1255	268	11	-	-	NOUN
ap-1255	268	12	expansions	expansion	NOUN
ap-1255	268	13	of	of	ADP
ap-1255	268	14	real	real	ADJ
ap-1255	268	15	numbers	number	NOUN
ap-1255	268	16	.	.	PUNCT
ap-1255	269	1	acta	acta	PROPN
ap-1255	269	2	math	math	PROPN
ap-1255	269	3	.	.	PUNCT
ap-1255	270	1	acad	acad	PROPN
ap-1255	270	2	.	.	PUNCT
ap-1255	271	1	sci	sci	PROPN
ap-1255	271	2	.	.	PUNCT
ap-1255	271	3	hungar	hungar	PROPN
ap-1255	271	4	.	.	PUNCT
ap-1255	272	1	11	11	NUM
ap-1255	272	2	,	,	PUNCT
ap-1255	272	3	1960	1960	NUM
ap-1255	272	4	,	,	PUNCT
ap-1255	272	5	401–416	401–416	NUM
ap-1255	272	6	.	.	PUNCT
ap-1255	273	1	[	[	X
ap-1255	273	2	8	8	NUM
ap-1255	273	3	]	]	X
ap-1255	273	4	praggastis	praggastis	PROPN
ap-1255	273	5	,	,	PUNCT
ap-1255	273	6	b.	b.	PROPN
ap-1255	273	7	:	:	PUNCT
ap-1255	273	8	markov	markov	PROPN
ap-1255	273	9	partitions	partition	NOUN
ap-1255	273	10	for	for	ADP
ap-1255	273	11	hyperbolic	hyperbolic	ADJ
ap-1255	273	12	toral	toral	ADJ
ap-1255	273	13	automorphisms	automorphism	NOUN
ap-1255	273	14	.	.	PUNCT
ap-1255	274	1	phd	phd	NOUN
ap-1255	274	2	thesis	thesis	NOUN
ap-1255	274	3	,	,	PUNCT
ap-1255	274	4	university	university	PROPN
ap-1255	274	5	of	of	ADP
ap-1255	274	6	washington	washington	PROPN
ap-1255	274	7	,	,	PUNCT
ap-1255	274	8	1994	1994	NUM
ap-1255	274	9	.	.	PUNCT
ap-1255	275	1	[	[	X
ap-1255	275	2	9	9	NUM
ap-1255	275	3	]	]	SYM
ap-1255	275	4	rauzy	rauzy	NOUN
ap-1255	275	5	,	,	PUNCT
ap-1255	275	6	g.	g.	PROPN
ap-1255	275	7	:	:	PUNCT
ap-1255	275	8	nombres	nombre	NOUN
ap-1255	275	9	algébriques	algébriques	PROPN
ap-1255	275	10	et	et	NOUN
ap-1255	275	11	substitutions	substitution	NOUN
ap-1255	275	12	.	.	PUNCT
ap-1255	276	1	bull	bull	NOUN
ap-1255	276	2	.	.	PUNCT
ap-1255	277	1	soc	soc	PROPN
ap-1255	277	2	.	.	PUNCT
ap-1255	278	1	math	math	PROPN
ap-1255	278	2	.	.	PUNCT
ap-1255	279	1	france	france	PROPN
ap-1255	279	2	,	,	PUNCT
ap-1255	279	3	110	110	NUM
ap-1255	279	4	,	,	PUNCT
ap-1255	279	5	1982	1982	NUM
ap-1255	279	6	,	,	PUNCT
ap-1255	279	7	147–178	147–178	NUM
ap-1255	279	8	.	.	PUNCT
ap-1255	280	1	[	[	X
ap-1255	280	2	10	10	NUM
ap-1255	280	3	]	]	SYM
ap-1255	280	4	rényi	rényi	NOUN
ap-1255	280	5	,	,	PUNCT
ap-1255	280	6	a.	a.	NOUN
ap-1255	280	7	:	:	PUNCT
ap-1255	280	8	representations	representation	NOUN
ap-1255	280	9	for	for	ADP
ap-1255	280	10	real	real	ADJ
ap-1255	280	11	numbers	number	NOUN
ap-1255	280	12	and	and	CCONJ
ap-1255	280	13	their	their	PRON
ap-1255	280	14	ergodic	ergodic	ADJ
ap-1255	280	15	properties	property	NOUN
ap-1255	280	16	.	.	PUNCT
ap-1255	281	1	acta	acta	PROPN
ap-1255	281	2	math	math	PROPN
ap-1255	281	3	.	.	PUNCT
ap-1255	282	1	acad	acad	PROPN
ap-1255	282	2	.	.	PUNCT
ap-1255	283	1	sci	sci	PROPN
ap-1255	283	2	.	.	PROPN
ap-1255	283	3	hungar	hungar	PROPN
ap-1255	283	4	,	,	PUNCT
ap-1255	283	5	8	8	NUM
ap-1255	283	6	,	,	PUNCT
ap-1255	283	7	1957	1957	NUM
ap-1255	283	8	,	,	PUNCT
ap-1255	283	9	477–493	477–493	NUM
ap-1255	283	10	.	.	PUNCT
ap-1255	284	1	[	[	X
ap-1255	284	2	11	11	NUM
ap-1255	284	3	]	]	SYM
ap-1255	284	4	thurston	thurston	PROPN
ap-1255	284	5	,	,	PUNCT
ap-1255	284	6	w.	w.	PROPN
ap-1255	284	7	p.	p.	PROPN
ap-1255	284	8	:	:	PUNCT
ap-1255	284	9	groups	group	NOUN
ap-1255	284	10	,	,	PUNCT
ap-1255	284	11	tilings	tiling	NOUN
ap-1255	284	12	,	,	PUNCT
ap-1255	284	13	and	and	CCONJ
ap-1255	284	14	finite	finite	VERB
ap-1255	284	15	state	state	NOUN
ap-1255	284	16	automata	automata	NOUN
ap-1255	284	17	.	.	PUNCT
ap-1255	285	1	ams	am	NOUN
ap-1255	285	2	colloquium	colloquium	NOUN
ap-1255	285	3	lecture	lecture	NOUN
ap-1255	285	4	notes	note	NOUN
ap-1255	285	5	,	,	PUNCT
ap-1255	285	6	1989	1989	NUM
ap-1255	285	7	.	.	PUNCT
ap-1255	286	1	ing	ing	PROPN
ap-1255	286	2	.	.	PUNCT
ap-1255	287	1	petr	petr	PROPN
ap-1255	287	2	ambrož	ambrož	PROPN
ap-1255	287	3	,	,	PUNCT
ap-1255	287	4	ph.d	ph.d	PROPN
ap-1255	287	5	.	.	PUNCT
ap-1255	288	1	e	e	X
ap-1255	288	2	-	-	NOUN
ap-1255	288	3	mail	mail	NOUN
ap-1255	288	4	:	:	PUNCT
ap-1255	288	5	petr.ambroz@fjfi.cvut.cz	petr.ambroz@fjfi.cvut.cz	PROPN
ap-1255	288	6	department	department	PROPN
ap-1255	288	7	of	of	ADP
ap-1255	288	8	mathematics	mathematics	PROPN
ap-1255	288	9	fnspe	fnspe	PROPN
ap-1255	288	10	,	,	PUNCT
ap-1255	288	11	czech	czech	PROPN
ap-1255	288	12	technical	technical	PROPN
ap-1255	288	13	university	university	PROPN
ap-1255	288	14	in	in	ADP
ap-1255	288	15	prague	prague	PROPN
ap-1255	288	16	trojanova	trojanova	X
ap-1255	288	17	13	13	NUM
ap-1255	288	18	,	,	PUNCT
ap-1255	288	19	120	120	NUM
ap-1255	288	20	00	00	NUM
ap-1255	288	21	praha	praha	PROPN
ap-1255	288	22	2	2	NUM
ap-1255	288	23	,	,	PUNCT
ap-1255	288	24	czech	czech	PROPN
ap-1255	288	25	republic	republic	NOUN
ap-1255	288	26	16	16	NUM