On uniformly distributed dilates of finite integer sequences (Q1568077)

Given \(N\) nonzero real numbers \(a_1<\cdots <a_N\) the authors consider the problem of finding a real number \(\alpha\) so that \(\alpha a_1, \dots, \alpha a_N\) are close to be uniformly distributed modulo one (this question is attributed to Komlos). First, it turns out that it suffices to consider integers \(a_1,\dots,a_N\). Given various quantities that measure how close a sequence is to being uniformly distributed, e.g., the size of the largest gap between consecutive points on the circle, discrepancy, or the number of points falling into any interval of size \(1/N\) (``concentration''), the authors provide upper bounds for the optimal dilate. These bounds depend only on \(N\) and they are attained by typical \(\alpha\), i.e., up to \(\alpha\) belonging to some set of small measure. Also lower bounds for these quantities are given. Some examples are constructed for this purpose by means of probabilistic methods. In case of the discrepancy, the lower and upper bounds match up to logarithms \((\sqrt{N/\log N} \) vs \( \sqrt N\log N)\). However, in case of the largest gap \((\log N/N \) vs \( N^{-1/2})\) and the concentration \((\exp(c \log N/\log \log^2N)\) vs \( N^{1/3+ \varepsilon})\) the lower and upper bounds do not match and the question about the correct asymptotic behaviour in terms of \(N\) remains open. Finally, the authors improve on a recent result of Noga Alon and the second author by showing that every set of \(N\) integers contains a non-averaging subset of size at least \(N^{1/5}\). The results are very interesting and give further insight into the uniform distribution properties of an important class of sequences.