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

Let (aₙ) ₍ ₍ be a Hadamard lacunary sequence. We give upper bounds for the maximal gap of the set of dilates \aₙ \₍ ₍ modulo 1, in terms of N. For any lacunary sequence (aₙ) ₍ ₍ we prove the existence of a dilation factor such that the maximal gap is of order at most (N) /N, and we prove that for Lebesgue almost all the maximal gap is of order at most (N) ^2+/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.