The dispersion of dilated lacunary sequences, with applications in multiplicative Diophantine approximation

Abstract

Let ${\left(a_n\right)}_{n\in N}$ be a lacunary sequence satisfying the Hadamard gap condition. We give upper bounds for the maximal gap of the set of dilates ${\left\{a_n\alpha \right\}}_{n\le N}$ modulo 1, in terms of N. For any lacunary sequence ${\left(a_n\right)}_{n\in N}$ we prove the existence of a dilation factor α such that the maximal gap is of order at most $(\log N)/N$, and we prove that for Lebesgue almost all α the maximal gap is of order at most ${(\log N)}^{2+\epsilon }/N$. The metric result is generalized to other measures satisfying a certain Fourier decay assumption. Both upper bounds are optimal up to a factor of logarithmic order, and the latter result improves a recent result of Chow and Technau. Finally, we show that our result implies an improved upper bound in the inhomogeneous version of Littlewood's problem in multiplicative Diophantine approximation.