Journal of the Australian Mathematical Society

Research Article

VARIATIONS AROUND A PROBLEM OF MAHLER AND MENDÈS FRANCE

YANN BUGEAUDa1

a1 Université de Strasbourg, Mathématiques, 7, rue René Descartes, 67084 Strasbourg, France (email: bugeaud@math.unistra.fr)

Abstract

We discuss the following general question and some of its extensions. Let (εk)k≥1 be a sequence with values in {0,1}, which is not ultimately periodic. Define ξ:=∑ k≥1εk/2k and ξ′:=∑ k≥1εk/3k. Let 𝒫 be a property valid for almost all real numbers. Is it true that at least one among ξ and ξ′ satisfies 𝒫?

(Received February 25 2011)

(Accepted January 16 2012)

2010 Mathematics subject classification

  • primary 11J82

Keywords and phrases

  • transcendental numbers;
  • algebraic numbers;
  • rational approximation;
  • numbers in different bases

Footnotes

Communicated by F. Pappalardi

To the memory of Alf van der Poorten