%0 Journal Article
%T On Liouville Sequences in the Non-Archimedean Case
%A Hamza Menken
%A Abdulkadir A£¿an
%J Chinese Journal of Mathematics
%D 2014
%R 10.1155/2014/813924
%X 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¨C8]). In 1975, Erd£¿s [9] proved a very interesting criterion for Liouville series. Theorem 1 (see Erd£¿s [9]). Let be an infinite sequence of integers satisfying for every and for fixed and . Then, is a Liouville number. Han£¿l [8] defined the concept of Liouville sequences and generalized the above theorem of Erd£¿s. 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
%U http://www.hindawi.com/journals/cjm/2014/813924/