On the maximal volume of empty convex bodies amidst multivariate dilates of a lacunary integer sequence

Let $(a_n{)}_{n\in \mathbb{N}}$ be a lacunary sequence of integers satisfying the Hadamard gap condition. For any fixed dimension $d\ge 1$, we establish asymptotic upper bounds for the maximal gap in the set of dilates $\{\mathit{\alpha }{a}_n{\}}_{n\le N}$ modulo 1 as $N\to \mathrm{\infty }$, for Lebesgue–almost all dilation vectors $\mathit{\alpha }\in [0,1{]}^d$. More precisely, we prove that for any lacunary $(a_n{)}_{n\in \mathbb{N}}$ and Lebesgue–almost all $\mathit{\alpha }$, every convex set in $[0,1{]}^d$ of volume at least $(\log N{)}^{2+\epsilon }\mathrm{/}N$ must contain an element of the set $\{\mathit{\alpha }{a}_n{\}}_{n\le N}$ mod 1, for all sufficiently large $N$. We also establish a generalized version of this result, where the $d$-dimensional Lebesgue measure is replaced by a general measure satisfying a certain Fourier decay condition. Our result is optimal up to logarithmic factors, and recovers as a special case a recent result for dimension $d=1$.