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 $d \geq 1$, we establish asymptotic upper bounds for the maximal gap in the set of dilates $\{\boldsymbol{\alpha} a_n \}_{n \leq N}$ modulo 1 as $N \to \infty$, for Lebesgue--almost all dilation vectors $\boldsymbol{\alpha} \in [0,1]^d$. More precisely, we prove that for any lacunary $(a_n)_{n \in \mathbb{N}}$ and Lebesgue--almost all $\boldsymbol{\alpha}$, every convex set in $[0,1]^d$ of volume at least $(\log N)^{2+\varepsilon}/N$ must contain an element of the set $\{\boldsymbol{\alpha} a_n \}_{n \leq N}$ 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 $d=1$.