On the Arithmetic Behavior of Liouville Numbers Under Rational Maps
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On the Arithmetic Behavior of Liouville Numbers Under Rational Maps Ana Paula Chaves1 · Diego Marques2 · Pavel Trojovský3 Received: 28 October 2019 / Accepted: 21 October 2020 © Sociedade Brasileira de Matemática 2020
Abstract In 1972, Alniaçik proved that every strong Liouville number is mapped into the set of Um -numbers, for any non-constant rational function with coefficients belonging to an m-degree number field. In this paper, we generalize this result by providing a larger class of Liouville numbers (which, in particular, contains the strong Liouville numbers) with this same property (this set is sharp is a certain sense). Keywords Mahler’s classification · Diophantine approximation · U -numbers · Rational functions Mathematics Subject Classification 11J81 · 11J82 · 11K60
1 Introduction The beginning of the transcendental number theory happened in 1844, when Liouville (1844) showed that algebraic numbers are not “well-approximated” by rationals. More precisely, if α is an n-degree real algebraic number (with n > 1), then there exists a positive constant C, depending only on α, such that |α − p/q| > Cq −n , for all rational number p/q. By using this result, Liouville was able to explicit, for the first time, examples of transcendental numbers (the now called Liouville numbers).
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Diego Marques [email protected] Ana Paula Chaves [email protected] Pavel Trojovský [email protected]
1
Instituto de Matemática e Estatística, Universidade Federal de Goiás, Goiânia, Brazil
2
Departamento de Matemática, Universidade de Brasília, Brasília, Brazil
3
Faculty of Science, University of Hradec Králové, Hradec Králové, Czech Republic
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A. P. Chaves et al.
A real number ξ is called a Liouville number if there exist infinitely many rational numbers ( pn /qn )n , with qn > 1, such that pn 1 < 0 < ξ − , qn qnωn for some sequence of real numbers (ωn )n which tends to ∞ as n → ∞. The set of the Liouville numbers is denoted by L. The first example of a Liouville number (and consequently, of a transcendental number) is the so-called Liouville constant defined by the convergent series = −n! (i.e., the decimal with 1’s in each factorial position and 0’s otherwise). 10 n≥1 Erdös (1962) proved that every real number can be written as the sum of two Liouville numbers (indeed, besides being a completely topological property of G δ -dense sets, Erdös was able to provide an explicit proof based only on the definition of Liouville numbers). Let ξ be a real irrational number and ( pk /qk )k be the sequence of the convergents of its continued fraction. It is well-known that the definition of Liouville numbers is equivalent to: for every positive integer n, there exist infinitely many positive integers k such that qk+1 > qkn . Thus, we have the following subclass of Liouville numbers. Definition 1 Let ξ be a real irrational number and ( pk /qk )k be the sequence of the convergents of its continued fraction. The number ξ is said to be a strong Liouville number if, for every n, there exists N (depending on n) su
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