EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 6, No. 2, 2013, 239-246
ISSN 1307-5543 – www.ejpam.com

On Some Properties of Liouville Numbers in the non-Archimedean
Case

Hamza Menken1,∗, Abdulkadir Aşan2

1 Mersin University, Science and Arts Faculty, Mathematics Department, Mersin-Turkey
2 Mersin University, Institute of Science, Mathematics Graduate Program, Mersin-Turkey

Abstract. We study Liouville numbers in the non-archimedean case. We give the analogues of the
Erdös theorem in the non-archimedean case, both in the p-adic numbers field Qp and the functions
field K 〈x〉.
2010 Mathematics Subject Classifications: 11J61, 12J25, 11R58
Key Words and Phrases: Non-archimedean field, p-adic number, p-adic Liouville number, functions
field.

1. Introduction

The classical Liouville’s theorem states that if α ∈ R is an algebraic number of degree
n≥ 2, then there exists a positive constant C(α) depending only on α such that

�

�

�α−
a

b

�

�

�≥
C(α)

bn

for all a, b ∈ Z+. The existence of transcendental numbers has been usually shown using the
Liouville’s theorem. For instance, the transcendence of the number ξ =

∑∞
n=1 10−n! can be

easily proved from the Liouville’s theorem [see 3]. A real number ξ ∈ R is called a (real)
Liouville number if for every positive integer n, there exist integer a and b(> 1) such that

0<
�

�

�ξ−
a

b

�

�

�<
1

bn .

Real Liouville numbers have many interesting properties and investigated by many authors
[see 2, 7, 9, 10, 12, 14]. We note that Liouville numbers are real numbers that can be rapidly
approximated by algebraic numbers with degree one. A general theory of approximation by
algebraic numbers is given in [5]. Here we mainly focus on the Erdös theorem:

∗Corresponding author.

Email addresses: hmenken@mersin.edu.tr (H. Menken), akadirasan@mersin.edu.tr (A. Aşan)

http://www.ejpam.com 239 c© 2013 EJPAM All rights reserved.



H. Menken, A. Aşan / Eur. J. Pure Appl. Math, 6 (2013), 239-246 240

Theorem 1. [P. Erdös, [8]] Let a1 < a2 < a3 < . . . be an infinite sequence of integers satisfying

lim
n→∞

sup a
1
tn
n =∞

for every t > 0, and
an > n1+ε

for fixed ε > 0 and n> n0 (ε). Then

α=
∞
∑

n=1

1

an

is a Liouville number.

It is well known that real numbers field R is archimedean. There are interesting non-
archimedean fields as the p-adic numbers field Qp and the functions field.

Let p be a fixed prime number. By Zp,Qp and Cp we denote the ring of p-adic integers,
the field of p-adic numbers, and the completion of the algebraic closure of Qp, respectively.

In the present work we investigate some properties of Liouville numbers in non-archimedean
case. Mainly, we give the analogues of the Erdös theorem in the non-archimedean case, both
in p-adic numbers field Qp and the functions field K 〈x〉.

Although the classical Liouville numbers are real numbers that can be rapidly approx-
imated by rational numbers, the p-adic Liouville numbers are those numbers that can be
rapidly approximated by positive integers in the p-adic norm. The p-adic Liouville numbers
are defined as follows:

Definition 1 ([6, 21]). Let α be a p-adic integer. If

lim
n→∞

inf n
p

|n−α|p = 0,

then the number α is called p-adic Liouville number.

Example 1. Let consider the series α =
∑∞

n=0 pn!. It is easy to see that the sum is a p-adic
Liouville number.

The definition above is first introduced by D. Clark [6] and it is better adapted to differ-
ential equations. In fact, consider the differential equation

x f ′(x)−λ f (x) =
1

1− x

on a neighborhood D of 0 in Zp where λ ∈ Zp\{0,1, 2, . . .}. This equation has an unique
formal solution, namely, f (x) =

∑∞
n=1

1
n−λ xn. It is clear that this solution divergent if only if

λ is a p-adic Liouville number (for details see [20]).
It is well known that the set L of p-adic Liouville numbers have the following basic

properties:

1. L ⊂ Zp



H. Menken, A. Aşan / Eur. J. Pure Appl. Math, 6 (2013), 239-246 241

2. L has measure 0 for the real Haar measure on Zp

3. If α ∈ L and n, m ∈ Z with m> 0, the n+mα ∈ L

4. L 6=−L and L ∩−L 6= ;

5. L forms a dense subset of Zp

6. Every α ∈ L is transcendental over Q.

In general case the p-adic transcendental numbers have been studied by K. Mahler [15],
W. W. Adams [1], X. X. Long [13], K. Nishioka [19] and others. As a special case the p-adic
Liouville numbers have been studied in [4, 11, 17, 18] and others.

2. The Erdös Theorem in the p-adic Numbers Field Qp.

We prove the following result as an analogue of the Erdös theorem in the p-adic numbers
field Qp.

Theorem 2. Let
�

an
�

be a sequence of p-adic integers such that

νp
�

an
�

< νp
�

an+1
�

(1)

for every n, and
νp
�

an+1
�

≥ n1+ε (2)

for fixed ε > 0 and n> n0 (ε). Then

α=
∞
∑

n=1

an

is a p-adic Liouville number.

Proof. First we show that the series
∑∞

n=1 an is convergent. It follows from the condition
(2) that

νp
�

an+1
�

≥ n1+ε

for fixed ε > 0 and n> n0 (ε). Then, we have
�

�an+1

�

�

p = p−νp(an+1) ≤ p−n1+ε
→ 0, (n→∞) .

Hence, lim
n→0

an = 0, so the series
∑∞

n=1 an is convergent. By the property
�

�

∑∞
n=1 an

�

�

p ≤max
n∈N

�

�an

�

�

p, we obtain that α=
∑∞

n=1 an ∈ Zp. Also, by the condition (1)

α ∈ Zp\Z.
Let ε > 0 be an arbitrary real number. Then,

0<
�

�α− Sn

�

�

1
n
p =

�

�

�

�

�

∞
∑

i=1

an+i

�

�

�

�

�

1
n

p

=
h

max
n

�

�an+1

�

�

p ,
�

�an+2

�

�

p , . . .
oi

1
n



H. Menken, A. Aşan / Eur. J. Pure Appl. Math, 6 (2013), 239-246 242

where Sn =
∑n

i=1 ai . Hence, from the condition (1) we obtain

0<
�

�α− Sn

�

�

1
n
p =

�

�an+1

�

�

1
n
p .

Thus,

0<
�

�α− Sn

�

�

1
n
p =

h

p−νp(an+1)
i

1
n

and by the inequality (2) we get

0<
�

�α− Sn

�

�

1
n
p =

h

p−νp(an+1)
i

1
n ≤ p−

n1+ε

n = p−nε �n≥ n0
�

.

Thus we have
�

�α− Sn

�

�

1
n
p → 0(n→∞).

Since Sn ∈ Zp for every n ∈ N, and the set of natural numbers N is dense in Zp, there exists a
sequence bn from N such that

�

�Sn− bn

�

�

p <
�

�α− Sn

�

�

p

for every n ∈ N. By the ultrametric inequality we can write

0<
�

�α− bn

�

�

p ≤max
n

�

�α− Sn

�

�

p ,
�

�Sn− bn

�

�

p

o

=
�

�α− Sn

�

�

p

for every n ∈ N. Hence, we can obtain a positive integer sequence bn such that

0<
�

�α− bn

�

�

1
n
p ≤

�

�α− Sn

�

�

1
n
p = p−nε → 0 (n→∞) .

So, the theorem is proved.

Remark 1. Since νp
�

an
�

∈ N for all an ∈ Zp, in Theorem 2, the condition (2) can be replaced
by the condition

νp
�

an+1
�

≥ n2.

In similar way, we can give the following result.

Corollary 1. Let
�

an
�

be a sequence of positive integers such that

νp
�

an
�

< νp
�

an+1
�

(3)

for every n, and
νp
�

an+1
�

≥ n2 (4)

for n> n0. Then

α=
∞
∑

n=1

an

is a p-adic Liouville number.



H. Menken, A. Aşan / Eur. J. Pure Appl. Math, 6 (2013), 239-246 243

Proof. By the relations (3) and (4) we have, l im
n→∞

an = 0, and so, the series
∑∞

n=1 an is

convergent and α ∈ Zp. Similarly, we can obtain that

0<
�

�α− Sn

�

�

1
n
p =

h

p−νp(an+1)
i

1
n ≤ p−

n2

n = p−n→ 0 (n→∞)

where Sn =
∑n

i=1 ai . Also, since Sn ∈ N for all n ∈ N, the number α is a p-adic Liouville
number.

3. The Erdös Theorem in the Functions Field K 〈x〉

Let K be an arbitrary field, x an indeterminate, K [x] the ring of all polynomials in x with
coefficients in K , K (x) the field of all rational functions in x with coefficients in K , and K 〈x〉
the field of all formal series

z = ak xk + ak−1 xk−1+ ak−2 xk−2+ . . .

in x where the coefficients ak, ak−1, ak−2, . . . are in K . Thus K (x) is the quotient field of K [x]
and a subfield of K 〈x〉.

A valuation |z| in K 〈x〉 is now defined by putting |0|= 0; but |z|= ek if

z = ak xk + ak−1 xk−1+ ak−2 xk−2+ . . .

and ak 6= 0.
If z lies in K [x], then log |z|= deg z.
It is clear that this norm is a non-archimedean and so, K 〈x〉 is a non-archimedean field

with this norm.
The analogue of Liouville’s theorem states that if α ∈ K 〈x〉 is an algebraic number of

degree n ≥ 2 over K(x), then there exists a positive constant C(α) depending only on α such
that

�

�

�α−
a

b

�

�

�≥
C(α)

bn

for all a, b ∈ K [x] (b 6= 0) [see 16]. Some investigations involve the Liouville numbers in the
functions field was done in [11]. Now we recall the definition of Liouville numbers in this
field.

Definition 2. An element ξ ∈ K 〈x〉 is called a Liouville number if for every ω ∈ R+, there exist
integer a, b ∈ K [x]\{0} with |b|> 1 such that

0<
�

�

�ξ−
a

b

�

�

�<
1

bω
.

We can give an analogue of the Erdös theorem in the functions field as follows



H. Menken, A. Aşan / Eur. J. Pure Appl. Math, 6 (2013), 239-246 244

Theorem 3. Let
�

zn
�

be a sequence of formal series in K 〈x〉 such that

deg
�

zn+1
�

< deg
�

zn
�

< 0 (5)

for every n and
deg
�

zn+1
�

≤−n1+ε (6)

for fixed ε > 0 and n> n0 (ε). Then,

α=
∞
∑

n=1

zn

is a Liouville number in K 〈x〉.

Proof. First we show that the series
∑∞

n=1 zn is convergent. It follows from the condition
(6) that

�

�zn+1

�

�= edeg(zn+1) ≤ e−n1+ε

for fixed ε > 0 and n> n0 (ε). Then, we get

l im
n→∞

zn = 0.

Thus, the series
∑∞

n=1 zn is convergent.
Let ε > 0 be an arbitrary real number. Then,

0<
�

�α− Sn

�

�

1
n =

�

�

�

�

�

∞
∑

i=1

zn+i

�

�

�

�

�

1
n

=
�

max
¦�

�zn+1

�

� ,
�

�an+2

�

� , . . .
©�

1
n

where Sn =
∑n

i=1 ai . Hence, from the condition (5) we obtain

0<
�

�α− Sn

�

�

1
n =
�

�zn+1

�

�

1
n .

Thus,

0<
�

�α− Sn

�

�

1
n =
h

edeg(zn+1)
i

1
n

and by the inequality (6) we get

0<
�

�α− Sn

�

�

1
n =
h

edeg(zn+1)
i

1
n ≤ e−

n1+ε

n = e−nε

for n> n0 (ε). Thus, we have
�

�α− Sn

�

�

1
n → 0(n→∞).

Since Sn ∈ K 〈x〉 for every n ∈ N, and the rational polynomials field set K(x) is dense in K 〈x〉
with respect the non-archimedean norm, there exists a sequence an

bn
∈ K(x) (an, bn ∈ K[x])

such that
�

�

�

�

Sn−
an

bn

�

�

�

�

<
�

�α− Sn

�

�



REFERENCES 245

for every n ∈ N. By the ultrametric inequality we can write
�

�

�

�

α−
an

bn

�

�

�

�

≤max
¦�

�α− Sn

�

� ,
�

�Sn− bn

�

�

©

=
�

�α− Sn

�

�

for every n ∈ N. Hence, we can obtain an
bn
∈ K(x) such that

�

�

�

�

α−
an

bn

�

�

�

�

1
n

≤
�

�α− Sn

�

�

1
n = e−nε → 0 (n→∞) .

So, α ∈ K 〈x〉 is a Liouville number.

Example 2. Consider the element ξ =
∑∞

n=1 x−n! in K 〈x〉. Let zn = x−n!. It is clear that zn
satisfy the conditions (5) and (6). By Theorem 3, ξ is a Liouville number in the functions field.

ACKNOWLEDGEMENTS This work is supported by Mersin University and the Scientific and
Technological Research Council of Turkey (TÜBİTAK). The authors would like to thank the
reviewers for their useful suggestions.

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