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

Let ( a n ) n is an element of N be a lacunary sequence satisfying the Hadamard gap condition. We give upper bounds for the maximal gap of the set of dilates { a n alpha } n <= N modulo 1, in terms of N . For any lacunary sequence ( a n ) n is an element of N we prove the existence of a dilation factor alpha such that the maximal gap is of order at most (log N ) /N , and we prove that for Lebesgue almost all alpha 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. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).