On uniform distribution of integers and uniform distribution mod. 1. (English) Zbl 0207.36101

The paper deals with the following conjecture of S. Uchiyama: Let \(\{b_n\}\) be a uniformly distributed sequence of integers, then the sequence \(\{b_nx\}\) is uniformly distributed mod 1 for almost all real numbers \(x\). The author shows that this conjecture is false. In fact a sequence of integers \(\{b_n\}\) is constructed such that \(\{b_n\}\) is uniformly distributed and \(\{b_nx\}\) is not uniformly distributed mod 1 for all \(x\) in a set \(V\) with \(V\subset (0,1)\) and Lebesgue measure \(V\ge \frac12\). The construction is based on some well-known results on continued fractions.
Reviewer: H. G. Meijer


11K06 General theory of distribution modulo \(1\)