  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.
