A common structure of \(n_k\)'s for which \(n_k\alpha\) mod \(1 \to x\) (Q2803022)

This paper shows an interesting new property of subsequences of \(\{n \alpha\}\)-sequences with given irrational number \(\alpha\). These subsequences satisfy \(\{n_{k}\alpha\}\rightarrow x\) for arbitrary given \(x\in [0,1]\) and \(\frac{k}{n_{k}}\geq \varepsilon_{k}\) for arbitrary given decreasing \(\varepsilon_{k}\rightarrow 0\). This complements work of \textit{A. Dubickas} [Proc. Am. Math. Soc. 137, No. 2, 449--456 (2009; Zbl 1221.11161)] and \textit{Y. Bugeaud} [Proc. Am. Math. Soc. 137, No. 8, 2609--2612 (2009; Zbl 1266.11084)]. An essential observation is that the differences \(n_{k+1}- n_{k}\) satisfy a certain version of the three gaps property.