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

Abstract

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