# On Liouville Sequences in the Non-Archimedean Case

Authors: Hamza Menken; Abdulkadir Aşan
Journal: Chinese Journal of Mathematics
 (2014)

---

## Abstract

We study Liouville numbers in the non-Archimedean case. We deal with the concept of a Liouville sequence in the non-Archimedean case and we give some results both in thep-adic numbers field ℚ
p and the functions field K
〈
x
〉.


---

## Body

## 1. Introduction

It is well known that if a complex numberα is a root of a nonzero polynomial equation
(1)
a
n
x
n
+
a
n
-
1
x
n
-
1
+
⋯
+
a
1
x
+
a
0
=
0
,
where the a
is are integers (or equivalently, rational numbers) and α satisfies no similar equation of degree <
n, then α is said to be an algebraic number of degree
n. A complex number that is not algebraic is said to be transcendental. Liouville’s theorem states that, for any algebraic number α with degree n
>
1, there exists C
(
α
)
>
0 such that
(2)
|
α
-
a
b
|
>
C
(
α
)
b
n
for all rational numbers a
/
b with b
>
1. The construction of transcendental numbers has been usually shown using Liouville's theorem. For instance, the transcendence of the number ξ
=
∑
n
=
1
∞
1
0
-
n
! can be easily proved from Liouville's theorem. Also, Liouville's theorem can be applied to prove the transcendence of a large class of real numbers which are called Liouville numbers.A real numberξ
∈
ℝ is called a Liouville number if, for every positive real number ω, there exist integers a and b
(
>
1
) such that
(3)
|
ξ
-
a
b
|
<
1
b
ω
.
It is easy to prove that any real number ∑
n
=
1
∞
a
n
1
0
-
n
! with a
n
∈
{
1,2
} is a Liouville number (see [1, 2]). Real Liouville numbers have many interesting properties and have been investigated by many authors (see [3–8]). In 1975, Erdös [9] proved a very interesting criterion for Liouville series.Theorem 1 (see Erdös [9]).
Leta
1
<
a
2
<
a
3
<
⋯ be an infinite sequence of integers satisfying
(4)
lim
⁡
n
→
∞
sup
⁡
a
n
t
1
/
n
=
∞
for every t
>
0 and
(5)
a
n
>
n
1
+
ɛ
for fixed ɛ
>
0 and n
>
n
0
(
ɛ
). Then,
(6)
α
=
∑
n
=
1
∞
‍
1
a
n
is a Liouville number.Hančl [8] defined the concept of Liouville sequences and generalized the above theorem of Erdös. Now, we recall the definition of Liouville sequences.Definition 2 (see [8]).
Let(
a
n
) be a sequence of positive real numbers. If, for every (
c
n
) of positive integers, the sum
(7)
∑
n
=
1
∞
‍
1
a
n
c
n
is a Liouville number, then the sequence (
a
n
) is called a Liouville sequence.The properties of Liouville sequences were investigated in [8] and some criteria were given for them. In the present work, we define the concept of Liouville sequences in non-Archimedean case and obtain some properties for them.


## 2.p-adic Numbers and p-adic Liouville Numbers

Recall that a norm on a fieldK is a function |
·
|
:
K
→
[
0
,
∞
) satisfying the following conditions:(i)
|
x
|
=
0 if and only if x
=
0,
(ii)
|
x
y
|
=
|
x
|
|
y
|, for all x
,
y
∈
K,
(iii)
|
x
+
y
|
≤
|
x
|
+
|
y
|, for all x
,
y
∈
K.A norm onK is called non-Archimedeanif it satisfies the extra condition(iv)
|
x
+
y
|
≤
max
⁡
{
|
x
|
,
|
y
|
} for all x
,
y
∈
K;otherwise, we say that the norm is Archimedean.It is well known that the usual absolute value on the rational numbers fieldℚ (or the real numbers field ℝ) is Archimedean. There are interesting non-Archimedean norms. First, we recall the definition of the p-adic norm.Letp be a given prime number. Every nonzero rational number x can be written uniquely under the form
(8)
x
=
p
v
p
(
x
)
a
b
,
where v
p
(
x
)
,
a
,
b
∈
ℤ, and a and b are not divided by p. Here, v
p
(
x
) is the p-adic valuation of x. The p-adic norm is defined by
(9)
|
x
|
p
=
{
p
-
v
p
(
x
)
,
x
≠
0
0
,
x
=
0
.
It is clear that the p-adic norm is non-Archimedean. The p-adic numbers field ℚ
p is the completion of the rational numbers field ℚ with respect to the p-adic norm. Every nonzero p-adic number x
∈
ℚ
p is uniquely represented in the canonical form
(10)
x
=
p
m
(
a
0
+
a
1
p
+
a
2
p
2
+
⋯
)
,
where m
=
v
p
(
x
)
,
a
j
∈
ℤ, such that 0
≤
a
j
≤
p
-
1 and a
0
≠
0 (j
=
0,1
,
2
,
…). The unit ball (or the ring of p-adic integers) is denoted by ℤ
p and defined by
(11)
ℤ
p
=
{
x
∈
ℚ
p
:
|
x
|
p
≤
1
}
.
Similarly, every nonzero p-adic integer x
∈
ℤ
p is uniquely represented in the canonical form
(12)
x
=
a
0
+
a
1
p
+
a
2
p
2
+
⋯
,
where a
j
∈
ℤ and 0
≤
a
j
≤
p
-
1  (
j
=
0,1
,
2
,
…). The natural numbers set ℕ is dense in ℤ
p.Although the classical Liouville numbers are real numbers that can be rapidly approximated by rational numbers, thep-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 3 (see [10, 11]).
Letλ be a p-adic integer. If
(13)
lim
⁡
n
→
∞
inf
⁡
|
n
-
λ
|
p
n
=
0
,
then λ is called a p-adic Liouville number.According to this definition,λ
∈
ℤ
p is a p-adic Liouville number if and only if there exists a sequence of positive integers a
n such that
(14)
lim
⁡
n
→
∞
|
a
n
-
λ
|
p
n
=
0
.Example 4.
Consider the seriesα
=
∑
n
=
0
∞
‍
p
n
!. It is easy to see that the sum is a p-adic Liouville number.The definition above is first introduced by Clark [11] and it is better adapted to differential equations. In fact, consider the differential equation
(15)
x
f
′
(
x
)
-
λ
f
(
x
)
=
1
1
-
x
on a neighborhood D of 0 in ℤ
p, where λ
∈
ℤ
p
∖
{
0,1
,
2
,
…
}. This equation has a unique formal solution; namely, f
(
x
)
=
∑
n
=
1
∞
‍
(
1
/
(
n
-
λ
)
)
x
n. It is clear that this solution diverges if and only if λ is a p-adic Liouville number (for details, see [12]). We note that the set of p-adic Liouville numbers forms a dense subset of ℤ
p and every p-adic Liouville number is transcendental over ℚ (for details, see [10]).In general case, thep-adic transcendental numbers have been studied by Mahler [13], Adams [14], X. Long Xin [15], Nishioka [16], and others. As a special case, the p-adic Liouville numbers have been studied in [17–21] and others.


## 3. Liouville Sequences in thep-adic Numbers Fields

We define the Liouville sequence inℚ
p as follows.Definition 5.
Let(
a
n
) be a sequence of p-adic integers. If, for every (
c
n
) of positive integers, the sum
(16)
∑
n
=
1
∞
‍
a
n
c
n
is a p-adic Liouville number, then the sequence (
a
n
) is called a p-adic Liouville sequence.Example 6.
Letp be a prime number. It is easy to see that
(17)
(
a
n
)
=
(
p
n
!
)
⊂
ℤ
p
is a p-adic Liouville sequence.Proof.
Let(
c
n
)
⊂
ℤ
+ be an arbitrary sequence. We want to show that the sum
(18)
γ
=
∑
n
=
1
∞
‍
c
n
p
n
!
is a p-adic Liouville number. Since |
c
n
p
n
!
|
p
≤
|
p
n
!
|
p
=
p
-
n
!
→
0, the series ∑
n
=
1
∞
‍
c
n
p
n
! is convergent. We can write
(19)
0
<
|
γ
-
S
n
-
1
|
p
1
/
n
=
|
∑
i
=
0
∞
‍
c
n
+
i
p
(
n
+
i
)
!
|
p
1
/
n
≤
[
max
⁡
{
|
c
n
p
n
!
|
p
,
|
c
n
+
1
p
(
n
+
1
)
!
|
p
,
…
}
]
1
/
n
=
[
|
c
n
|
p
|
p
n
!
|
p
]
1
/
n
,
where S
n
=
∑
k
=
1
n
‍
c
k
p
k
!. Since c
n is an integer, then we get |
c
n
|
p
≤
1 and
(20)
0
<
|
γ
-
S
n
-
1
|
p
1
/
n
≤
|
p
n
!
|
p
1
/
n
=
p
-
(
n
-
1
)
!
.
Thus,
(21)
lim
⁡
n
→
∞
|
γ
-
S
n
-
1
|
p
1
/
n
=
lim
⁡
n
→
∞
p
-
(
n
-
1
)
!
=
0
.
This shows that the sum
(22)
γ
=
∑
n
=
1
∞
‍
c
n
p
n
!
is a p-adic Liouville number.Theorem 7.
Let(
a
n
) be a sequence of positive integers satisfying the following conditions:
(23)
ν
p
(
a
n
)
<
ν
p
(
a
n
+
1
)
,
for every n, and
(24)
ν
p
(
a
n
+
1
)
≥
n
1
+
ɛ
for fixed ɛ
>
0 and n
>
n
0
(
ɛ
). Then, (
a
n
) is a p-adic Liouville sequence.Proof.
Let(
c
n
) be an arbitrary sequence of positive integers and let ɛ
>
0 be a given arbitrary positive real number. First, we have to prove that the series ∑
n
=
1
∞
‍
a
n
c
n is convergent. By condition (24), we know that
(25)
ν
p
(
a
n
+
1
)
≥
n
1
+
ɛ
,
for all n
>
n
0
(
ɛ
). It follows from |
c
n
|
p
≤
1 that the relation holds
(26)
|
a
n
+
1
c
n
+
1
|
p
≤
|
a
n
+
1
|
p
=
p
-
ν
p
(
a
n
+
1
)
≤
p
-
n
1
+
ɛ
,
for all n
>
n
0
(
ɛ
). Thus, lim
⁡
n
→
0
c
n
a
n
=
0, so the series ∑
n
=
1
∞
‍
c
n
a
n is convergent. By the property |
∑
n
=
1
∞
‍
c
n
a
n
|
p
≤
max
⁡
n
∈
ℕ
|
c
n
a
n
|
p, we obtain that α
=
∑
n
=
1
∞
‍
c
n
a
n
∈
ℤ
p. Also, by condition (23), α
∈
ℤ
p
∖
ℤ. Now, we want to show that the sum
(27)
α
=
∑
n
=
1
∞
‍
a
n
c
n
is a p-adic Liouville number. Using (23) and (24), we have
(28)
0
<
|
α
-
S
n
|
p
1
/
n
=
|
∑
i
=
1
∞
‍
a
n
+
i
c
n
+
i
|
p
1
/
n
≤
[
max
⁡
{
|
a
n
+
1
c
n
+
1
|
p
,
|
a
n
+
2
c
n
+
2
|
p
,
…
}
]
1
/
n
=
|
a
n
+
1
c
n
+
1
|
p
1
/
n
,
where S
n
=
∑
k
=
1
n
‍
a
k
c
k. Since |
c
n
+
i
|
p
≤
1, for all i
=
1,2
,
3
,
…, then
(29)
0
<
|
α
-
S
n
|
p
1
/
n
=
|
∑
i
=
1
∞
‍
a
n
+
i
c
n
+
i
|
p
1
/
n
≤
|
a
n
+
1
|
p
1
/
n
=
p
-
υ
p
(
a
n
+
1
)
/
n
≤
p
-
n
1
+
ɛ
/
n
=
p
-
n
ɛ
⟶
0
(
n
⟶
∞
)
.
Since S
n
∈
ℕ, for all n, this shows that α is a p-adic Liouville number and the theorem is proved.Remark 8.
Sinceν
p
(
a
n
)
∈
ℕ, for all a
n
∈
ℤ
p, in Theorem 7, condition (24) can be replaced by the condition
(30)
ν
p
(
a
n
+
1
)
≥
n
2
.In similar way, we can give the following result.Corollary 9.
Let(
a
n
) be a sequence of p-adic integers satisfying the following conditions:
(31)
ν
p
(
a
n
)
<
ν
p
(
a
n
+
1
)
,
for every n, and
(32)
ν
p
(
a
n
+
1
)
≥
n
2
,
for n
>
n
0. Then, (
a
n
) is a p-adic Liouville sequence.Theorem 10.
Let(
a
n
) be a sequence of positive integers and assume that the relation
(33)
0
<
|
a
n
|
p
1
/
n
<
ϵ
holds for every positive real number ϵ
>
0 and n
>
n
0
(
ɛ
). Then, (a)
α
=
∑
n
=
1
∞
‍
a
n is a p-adic Liouville number,
(b)
(
a
n
) is a p-adic Liouville sequence.Proof.
Letϵ
>
0 be a given arbitrary positive real number.
(a) By condition (33), there exists n
0
(
ɛ
)
∈
ℕ such that the relation
(34)
0
<
|
a
n
|
p
1
/
n
<
ϵ
holds, for all n
>
n
0
(
ɛ
). Then, we get |
a
n
|
p
<
ϵ
n and |
a
n
|
p
→
0 as n
→
∞. Hence, the series ∑
n
=
1
∞
‍
a
n is convergent and by the inequality
(35)
|
∑
n
=
1
∞
‍
a
n
|
p
≤
max
⁡
n
∈
ℕ
|
a
n
|
p
≤
1
,
we have α
=
∑
n
=
1
∞
‍
a
n
∈
ℤ
p. Now, we show that α
=
∑
n
=
1
∞
‍
a
n  is a
p
-adic Liouville number. Let
α
n
=
∑
k
=
1
n
‍
a
k. Then, we can write
(36)
0
<
|
α
-
α
n
|
p
1
/
n
=
|
∑
k
=
n
+
1
∞
‍
a
k
|
p
1
/
n
≤
max
⁡
k
≥
n
+
1
|
a
k
|
p
1
/
k
<
ϵ
,
for all n
>
n
0
(
ɛ
). It follows that
(37)
|
α
-
α
n
|
p
1
/
n
⟶
0
.
Hence, we obtain that α
=
∑
n
=
1
∞
‍
a
n is a p-adic Liouville number.
(b) Let(
c
n
) be an arbitrary sequence of positive integers. We consider the sum:
(38)
β
=
∑
n
=
1
∞
‍
c
n
a
n
.
We know that the relation
(39)
0
<
|
a
n
|
p
1
/
n
<
ϵ
holds, for all n
>
n
0
(
ɛ
). Since |
c
n
|
p
≤
1, we get
(40)
|
a
n
c
n
|
p
≤
|
a
n
|
p
<
ϵ
n
and |
a
n
c
n
|
p
→
0 as n
→
∞. Hence, the series ∑
n
=
1
∞
‍
a
n
c
n is convergent and by the inequality
(41)
|
∑
n
=
1
∞
‍
a
n
c
n
|
p
≤
max
⁡
n
∈
ℕ
|
a
n
c
n
|
p
≤
1
,
we have β
=
∑
n
=
1
∞
‍
a
n
c
n
∈
ℤ
p. Let  β
n
=
∑
k
=
1
n
‍
c
k
a
k. Then, we can write
(42)
0
<
|
β
-
β
n
|
p
1
/
n
=
|
∑
k
=
n
+
1
∞
‍
c
k
a
k
|
p
1
/
n
≤
max
⁡
k
≥
n
+
1
|
c
k
a
k
|
p
1
/
n
≤
max
⁡
k
≥
n
+
1
|
a
k
|
p
1
/
n
<
ϵ
for all n
>
n
0
(
ɛ
). It follows that
(43)
|
β
-
β
n
|
p
1
/
n
⟶
0
.
Since β
n
∈
ℕ, for all n, we obtain that β
=
∑
n
=
1
∞
‍
c
n
a
n is a p-adic Liouville number. So, the theorem is proved.


## 4. The Liouville Sequences in the Functions Field

LetK 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
(44)
z
=
a
k
x
k
+
a
k
-
1
x
k
-
1
+
a
k
-
2
x
k
-
2
+
⋯
in x, where the coefficients a
k
,
a
k
-
1
,
a
k
-
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 and |
z
|
=
e
k if
(45)
z
=
a
k
x
k
+
a
k
-
1
x
k
-
1
+
a
k
-
2
x
k
-
2
+
⋯
and a
k
≠
0.Ifz lies in K
[
x
], then log
⁡
|
z
|
=
deg
⁡
z.It is clear that this norm is a non-Archimedean and soK
〈
x
〉 is a non-Archimedean field with this norm.The analogue of Liouville's theorem states that ifα
∈
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
(46)
|
α
-
a
b
|
≥
C
(
ξ
)
b
n
for all a
,
b
∈
K
[
x
]
(
b
≠
0
) (see [22]). Some results on the Liouville numbers in the functions field were obtained in [20]. Now, we recall the definition of a Liouville number in this field.Definition 11.
An elementξ
∈
K
〈
x
〉 is called aLiouville number if, for every ω
∈
ℝ
+, there existed integer a
,
b
∈
K
[
x
]
∖
{
0
} with |
b
|
>
1 such that
(47)
0
<
|
ξ
-
a
b
|
<
1
b
ω
.We define the concept of a Liouville sequence in the function fields as follows.Definition 12.
Let(
z
n
)
⊂
K
〈
x
〉. If, for every (
a
n
)
∈
K
[
x
], the sum
(48)
∑
n
=
1
∞
‍
1
a
n
z
n
is a Liouville number, then the sequence (
z
n
) is called a Liouville sequence.Theorem 13.
Let(
z
n
)
⊂
K
〈
x
〉 satisfying the following conditions:
(49)
0
<
deg
⁡
(
z
n
)
<
deg
⁡
(
z
n
+
1
)
for every n, and
(50)
deg
⁡
(
z
n
+
1
)
>
n
1
+
ε
,
for fixed ɛ
>
0 and n
>
n
0
(
ɛ
). Then, (
z
n
) is a Liouville sequence.Proof.
Let(
a
n
)
⊂
K
[
x
] be an arbitrary sequence and ɛ
>
0 be an arbitrary positive real number. First, we show that ∑
n
=
1
∞
‍
(
1
/
a
n
z
n
) is convergent in K
〈
x
〉. From condition (50), we have
(51)
|
1
a
n
z
n
|
=
e
-
deg
⁡
(
a
n
z
n
)
≤
e
-
deg
⁡
(
z
n
)
<
e
-
(
n
-
1
)
1
+
ɛ
for all n
>
n
0
(
ɛ
). Then, we get 1
/
a
n
z
n
→
0. Thus, the series ∑
n
=
1
∞
‍
(
1
/
a
n
z
n
) is convergent. Now, we want to show that the sum
(52)
α
=
∑
n
=
1
∞
‍
1
a
n
z
n
is a Liouville number. Let S
n
=
∑
k
=
1
n
‍
(
1
/
a
k
z
k
). Then, we write
(53)
0
<
|
α
-
S
n
|
1
/
n
=
|
∑
i
=
1
∞
‍
1
a
n
+
i
z
n
+
i
|
1
/
n
≤
[
max
⁡
k
≥
n
+
1
1
|
a
k
z
k
|
]
1
/
n
.
Since deg
⁡
(
a
k
)
≥
0, we obtain that
(54)
[
max
⁡
k
≥
n
+
1
1
|
a
k
z
k
|
]
1
/
n
=
[
max
⁡
k
≥
n
+
1
e
-
deg
⁡
(
a
k
)
-
deg
⁡
(
z
k
)
]
1
/
n
≤
[
max
⁡
k
≥
n
+
1
e
-
deg
⁡
(
z
k
)
]
1
/
n
.
By (49), we can write
(55)
max
⁡
k
≥
n
+
1
e
-
deg
⁡
(
z
k
)
=
e
-
deg
⁡
(
z
n
+
1
)
,
and by using (50) we get
(56)
[
max
⁡
k
≥
n
+
1
1
|
a
k
z
k
|
]
1
/
n
≤
[
max
⁡
k
≥
n
+
1
e
-
deg
⁡
(
z
k
)
]
1
/
n
=
e
-
deg
⁡
(
z
n
+
1
)
/
n
≤
e
-
n
1
+
ɛ
/
n
=
e
-
n
ɛ
⟶
0
(
n
⟶
∞
)
.
This shows that|
α
-
S
n
|
1
/
n
→
0
,
n
→
∞. Also, by condition (49), z
n
∈
K
[
x
] and so S
n
∈
K
(
x
). Thus, we prove that α is a Liouville number.



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*Source: 813924-2014-03-05.xml*