Chinese Journal of Mathematics

Volume 2014 (2014), Article ID 813924, 6 pages

http://dx.doi.org/10.1155/2014/813924

Research Article

On Liouville Sequences in the Non-Archimedean Case

Hamza Menken¹ and Abdulkadir Aşan²

¹Mathematics Department, Science and Arts Faculty, Mersin University, Ciftlikkoy Campus, 33343 Mersin, Turkey

²Institute of Science, Mathematics Graduate Program, Mersin University, Ciftlikkoy Campus, 33343 Mersin, Turkey

Received 16 December 2013; Accepted 27 January 2014; Published 5 March 2014

Academic Editors: P. K. Sahoo, H. Wu, and Z. Wu

Copyright © 2014 Hamza Menken and Abdulkadir Aşan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 the p-adic numbers field and the functions field .

1. 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 , there exists such that

for all rational numbers with . The construction of transcendental numbers has been usually shown using Liouville's theorem. For instance, the transcendence of the number 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 number is called a Liouville number if, for every positive real number , there exist integers and such that

It is easy to prove that any real number with 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]). Let be an infinite sequence of integers satisfying

for every and

for fixed and . Then,

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 be a sequence of positive real numbers. If, for every of positive integers, the sum

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.

2. -adic Numbers and -adic Liouville Numbers

Recall that a norm on a field is a function satisfying the following conditions:

(i) if and only if ,(ii), for all ,(iii), for all .

A norm on is called non-Archimedean if it satisfies the extra condition

(iv) for all ;

otherwise, we say that the norm is Archimedean.

It is well known that the usual absolute value on the rational numbers field (or the real numbers field ) 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 , and and are not divided by . Here, 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 is the completion of the rational numbers field with respect to the -adic norm. Every nonzero -adic number is uniquely represented in the canonical form

where , such that and (). The unit ball (or the ring of -adic integers) is denoted by and defined by

Similarly, every nonzero -adic integer is uniquely represented in the canonical form

where and   ). The natural numbers set is dense in .

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.

Definition 3 (see [10, 11]). Let be a -adic integer. If

then is called a -adic Liouville number.

According to this definition, is a -adic Liouville number if and only if there exists a sequence of positive integers such that

Example 4. Consider the series . 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 in , where . This equation has a unique formal solution; namely, . 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 and every -adic Liouville number is transcendental over (for details, see [10]).

In general case, the -adic transcendental numbers have been studied by Mahler [13], Adams [14], X. Long Xin [15], Nishioka [16], and others. As a special case, the -adic Liouville numbers have been studied in [17–21] and others.

3. Liouville Sequences in the -adic Numbers Fields

We define the Liouville sequence in 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 be an arbitrary sequence. We want to show that the sum

is a -adic Liouville number. Since , the series is convergent. We can write

where . Since is an integer, then we get and

Thus,

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 and . Then, is a -adic Liouville sequence.

Proof. Let be an arbitrary sequence of positive integers and let be a given arbitrary positive real number. First, we have to prove that the series is convergent. By condition (24), we know that

for all . It follows from that the relation holds

for all . Thus, , so the series is convergent. By the property , we obtain that . Also, by condition (23), . Now, we want to show that the sum

is a -adic Liouville number. Using (23) and (24), we have

where . Since , for all , then

Since , for all , this shows that is a -adic Liouville number and the theorem is proved.

Remark 8. Since , for all , 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 . Then, is a -adic Liouville sequence.

Theorem 10. Let be a sequence of positive integers and assume that the relation

holds for every positive real number and . Then,

(a) is a -adic Liouville number,(b) is a -adic Liouville sequence.

Proof. Let be a given arbitrary positive real number.

(a) By condition (33), there exists such that the relation

holds, for all . Then, we get and as . Hence, the series is convergent and by the inequality

we have . Now, we show that   is a -adic Liouville number. Let . Then, we can write

for all . It follows that

Hence, we obtain that is a -adic Liouville number.

(b) Let be an arbitrary sequence of positive integers. We consider the sum:

We know that the relation

holds, for all . Since , we get

and as . Hence, the series is convergent and by the inequality

we have . Let  . Then, we can write

for all . It follows that

Since , for all , we obtain that is a -adic Liouville number. So, the theorem is proved.

4. 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

in , where the coefficients are in . Thus, is the quotient field of and a subfield of .

A valuation in is now defined by putting and if

and .

If lies in , then .

It is clear that this norm is a non-Archimedean and so is a non-Archimedean field with this norm.

The analogue of Liouville's theorem states that if is an algebraic number of degree over , then there exists a positive constant depending only on such that

for all (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 element is called a Liouville number if, for every , there existed integer with such that

We define the concept of a Liouville sequence in the function fields as follows.

Definition 12. Let . If, for every , the sum

is a Liouville number, then the sequence is called a Liouville sequence.

Theorem 13. Let satisfying the following conditions:

for every , and

for fixed and . Then, is a Liouville sequence.

Proof. Let be an arbitrary sequence and be an arbitrary positive real number. First, we show that is convergent in . From condition (50), we have

for all . Then, we get . Thus, the series is convergent. Now, we want to show that the sum

is a Liouville number. Let . Then, we write

Since , we obtain that

By (49), we can write

and by using (50) we get

This shows that . Also, by condition (49), and so . Thus, we prove that is a Liouville number.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by Mersin University. The authors would like to thank the editors and reviewers for their useful suggestions.

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