Research Article | Open Access
On Liouville Sequences in the Non-Archimedean Case
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 .
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  proved a very interesting criterion for Liouville series.
Theorem 1 (see Erdös ). Let be an infinite sequence of integers satisfying for every and for fixed and . Then, is a Liouville number.
Hančl  defined the concept of Liouville sequences and generalized the above theorem of Erdös. Now, we recall the definition of Liouville sequences.
Definition 2 (see ). 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  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.
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  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 ). 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 ).
In general case, the -adic transcendental numbers have been studied by Mahler , Adams , X. Long Xin , Nishioka , 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.
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 ). Some results on the Liouville numbers in the functions field were obtained in . 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.
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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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.