Meijer, H. G. On uniform distribution of integers and uniform distribution mod. 1. (English) Zbl 0207.36101 Nieuw Arch. Wiskd., III. Ser. 18, 271-278 (1970). 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 Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 2 Documents MSC: 11K06 General theory of distribution modulo \(1\) Keywords:uniform distribution of integers; uniform distribution modulo one PDF BibTeX XML Cite \textit{H. G. Meijer}, Nieuw Arch. Wiskd., III. Ser. 18, 271--278 (1970; Zbl 0207.36101)