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, 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
References
[1]
A. Baker, Number Theory, Cambridge University Press, Cambridge, UK, 1975.
[2]
Y. Bugeaud, Approximation by Algebraic Numbers, Cambridge University Press, Cambridge, UK, 2004.
[3]
K. Mahler, “Zur Approximation der Exponentialfunktion und des Logarithmus I, II,” Journal für die Reine und Angewandte Mathematik, vol. 166, pp. 118–150, 1932.
[4]
J. F. Koksma, “über die Mahlersche Klasseneinteilung der transzendenten Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen,” Monatshefte für Mathematik und Physik, vol. 48, no. 1, pp. 176–189, 1939.
[5]
P. Erd？s, “Representation of real numbers as sums and products of Liouville numbers,” Michigan Mathematical Journal, vol. 9, pp. 59–60, 1962.
[6]
W. J. Leveque, “On Mahler's U-Numbers,” London Mathematical Society, pp. 220–229, 1953.
[7]
K. Alnia？ik, “On Um-numbers,” Proceedings of the American Mathematical Society, vol. 85, no. 4, pp. 499–505, 1982.
[8]
J. Han？l, “Liouville sequences,” Nagoya Mathematical Journal, vol. 172, pp. 173–187, 2003.
[9]
P. Erd？s, “Some problems and results on the irrationality of the sum of infinite series,” Michigan Mathematical Journal, pp. 1–7, 1975.
[10]
W. H. Schikhof, Ultrametric Calculus, Cambridge University press, Cambridge, UK, 2006.
[11]
D. Clark, “A note on the p-adic convergence of the solutions of linear differential equations,” Proceedings of the American Mathematical Society, vol. 17, pp. 262–269, 1966.
[12]
M. van der Put and L. Taelman, “Local p-adic differential equations, p-Adic Mathematical Physics,” in AIP Conference Proceedings, vol. 826, pp. 291–297, The American Institute of Physics, Melville, NY, USA, 2006.
[13]
K. Mahler, “Uber eine Klassen-Einteilung der p-adischen Zahlen,” Mathematica, vol. 3, pp. 177–185, 1935.
[14]
W. W. Adams, “Transcendental numbers in the p-adic domain,” The American Journal of Mathematics, vol. 88, pp. 279–308, 1966.
[15]
X. Long Xin, “Mahler's Classification of p-adic numbers,” Pure and Applied Mathematics, vol. 5, pp. 73–780, 1989.
[16]
K. Nishioka, “p-adic transcendental numbers,,” Proceedings of the American Mathematical Society, vol. 108, no. 1, pp. 39–41, 1990.
[17]
H. Menken, “An investigation on p-adic U-number,” Istanbul Universitesi Fen Fakültesi Matematik Dergisi, vol. 59, pp. 111–143, 2001.
[18]
V. V. Beresnevich, V. I. Bernik, and E. I. Kovalevskaya, “On approximation of p-adic numbers by p-adic algebraic numbers,” Journal of Number Theory, vol. 111, no. 1, pp. 33–56, 2005.
[19]
H. Menken and K. R. Mamedov, “Point on curves whose coordinates are p-adic U-numbers,” in Proceedings of the 2nd International Conference on p-Adic Mathematical Physics, vol. 826 of AIP Conference Proceedings, pp. 267–273, Melville, NY, USA, September 2006.
[20]
T. Chaichana, T. Komatsu, and V. Laohakosol, “Liouville numbers in the non-archimedean case,” Publicationes Mathematicae, vol. 77, no. 1-2, pp. 39–63, 2010.
[21]
H. Menken and A. A？an, “On some properties of Liouville numbers in the non-Archimedean case,” European Journal of Pure and Applied Mathematics, vol. 6, no. 2, pp. 239–246, 2013.
[22]
K. Mahler, “On a theorem of Liouville in fields of positive characteristic,” Canadian Journal of Mathematics, vol. 1, pp. 397–400, 1949.
