On Liouville Sequences in the Non-Archimedean Case

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 10 −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 numberswhich are called Liouville numbers. A real number ξ ∈ R is called a Liouville number if, for every positive real numberω, there exist integers a and b(> 1) such that 󵄨󵄨󵄨󵄨󵄨󵄨 ξ − a


Introduction
It is well known that if a complex number  is a root of a nonzero polynomial equation where the   s are integers (or equivalently, rational numbers) and  satisfies no similar equation of degree < , then  is said to be an algebraic number of degree .A complex number that is not algebraic is said to be transcendental.Liouville's theorem states that, for any algebraic number  with degree  > 1, there exists () > 0 such that for all rational numbers / with  > 1.The construction of transcendental numbers has been usually shown using Liouville's theorem.For instance, the transcendence of the number  = ∑ ∞ =1 10 −! 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.
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 (  ) be a sequence of positive real numbers.If, for every (  ) of positive integers, the sum Chinese Journal of Mathematics is a Liouville number, then the sequence (  ) 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.

Liouville Numbers
Recall that a norm on a field  is a function |⋅| :  → [0, ∞) satisfying the following conditions: It is well known that the usual absolute value on the rational numbers field Q (or the real numbers field R) is Archimedean.There are interesting non-Archimedean norms.First, we recall the definition of the -adic norm.
Let  be a given prime number.Every nonzero rational number  can be written uniquely under the form where V  (), ,  ∈ Z, and  and  are not divided by .Here, V  () is the -adic valuation of .The -adic norm is defined by It is clear that the -adic norm is non-Archimedean.The -adic numbers field Q  is the completion of the rational numbers field Q with respect to the -adic norm.Every nonzero -adic number  ∈ Q  is uniquely represented in the canonical form where  = V  (),   ∈ Z, such that 0 ≤   ≤  − 1 and  0 ̸ = 0 ( = 0, 1, 2, . ..).The unit ball (or the ring of -adic integers) is denoted by Z  and defined by Similarly, every nonzero -adic integer  ∈ Z  is uniquely represented in the canonical form where   ∈ Z and 0 ≤   ≤  − 1 ( = 0, 1, 2, . ..).The natural numbers set N is dense in Z  .
Although the classical Liouville numbers are real numbers that can be rapidly approximated by rational numbers, the -adic Liouville numbers are those numbers that can be rapidly approximated by positive integers in the -adic norm.The -adic Liouville numbers are defined as follows.
According to this definition,  ∈ Z  is a -adic Liouville number if and only if there exists a sequence of positive integers   such that lim Example 4. Consider the series  = ∑ ∞ =0  ! .It is easy to see that the sum is a -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 on a neighborhood  of 0 in Z  , where  ∈ Z  \ {0, 1, 2, . ..}.This equation has a unique formal solution; namely, () = ∑ ∞ =1 (1/( − ))  .It is clear that this solution diverges if and only if  is a -adic Liouville number (for details, see [12]).We note that the set of -adic Liouville numbers forms a dense subset of Z  and every -adic Liouville number is transcendental over Q (for details, see [10]).

Liouville Sequences in the 𝑝-adic Numbers Fields
We define the Liouville sequence in Q  as follows.
Definition 5. Let (  ) be a sequence of -adic integers.If, for every (  ) of positive integers, the sum is a -adic Liouville number, then the sequence (  ) is called a -adic Liouville sequence.
Example 6.Let  be a prime number.It is easy to see that is a -adic Liouville sequence.
Proof.Let (  ) ⊂ Z + be an arbitrary sequence.We want to show that the sum where   = ∑  =1    ! .Since   is an integer, then we get Thus, lim This shows that the sum is a -adic Liouville number.
Theorem 7. Let (  ) be a sequence of positive integers satisfying the following conditions: for every , and for fixed  > 0 and  >  0 ().Then, (  ) is a -adic Liouville sequence.
Proof.Let (  ) be an arbitrary sequence of positive integers and let  > 0 be a given arbitrary positive real number.First, we have to prove that the series ∑ ∞ =1     is convergent.By condition (24), we know that for all  >  0 ().It follows from |  |  ≤ 1 that the relation holds for all  >  0 ().Thus, lim  → 0     = 0, so the series ∑ Since   ∈ N, for all , this shows that  is a -adic Liouville number and the theorem is proved.
Remark 8. Since ]  (  ) ∈ N, for all   ∈ Z  , in Theorem 7, condition (24) can be replaced by the condition In similar way, we can give the following result.
Corollary 9. Let (  ) be a sequence of -adic integers satisfying the following conditions: for every , and for  >  0 .Then, (  ) is a -adic Liouville sequence.
(b) Let (  ) be an arbitrary sequence of positive integers.We consider the sum: We know that the relation 0 < Since   ∈ N, for all , we obtain that  = ∑ ∞ =1     is a -adic Liouville number.So, the theorem is proved.

The Liouville Sequences in the Functions Field
Let  be an arbitrary field,  an indeterminate, [] the ring of all polynomials in  with coefficients in , () the field of all rational functions in  with coefficients in , and ⟨⟩ the field of all formal series for all ,  ∈ [] ( ̸ = 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.(56) This shows that | −   | 1/ → 0,  → ∞.Also, by condition (49),   ∈ [] and so   ∈ ().Thus, we prove that  is a Liouville number.