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

Let $(a_n)_{n \in \mathbb{N}}$ be a Hadamard lacunary sequence. We give upper bounds for the maximal gap of the set of dilates $\{a_n \alpha\}_{n \leq N}$ modulo 1, in terms of $N$. For any lacunary sequence $(a_n)_{n \in \mathbb{N}}$ 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 $(\log N)^{2+\varepsilon}/N$. 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.