On Uniformly Distributed Dilates of Finite Integer Sequences

Given N nonzero real numbers a1<…<aN, we consider the problem of finding a real number α so that αa1, …, αaN are close to be uniformly distributed modulo one (this question is attributed to Komlos). First, it turns out that it suffices to consider integers a1, …, aN. 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”), we provide upper bounds for the optimal dilate. These bounds depend only on N and they are attained by typical α, i.e., up to α belonging to some set of small measure. We also provide lower bounds for these quantities. Some of our 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 (N/logN vs 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 log2 N) vs N1/3+e) the lower and upper bounds do not match and the question about the correct asymptotic behavior in terms of N remains open. Finally, we 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 N1/5.