On the maximal volume of empty convex bodies amidst multivariate dilates of a lacunary integer sequence
Year: 2025
Type: preprint
Abstract: 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 \(\{\boldsymbolα a_n \}_{n \leq N}\) modulo 1 as $N \to \infty$, for Lebesgue--almost all dilation vectors $\boldsymbolα \in [0,1]^d$. More precisely, we prove that for any lacunary \((a_n)_{n \in \mathbb{N}}\) and Lebesgue--almost all $\boldsymbolα$, every convex set in $[0,1]^d$ of volume at least $(\log N)^{2+\varepsilon}/N$ 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 $d=1$.
Source: arXiv (Cornell University)
Author: Stefanescu, Eduard
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6979357122
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