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

2025/04/04 by Stefanescu, Eduard
#11J70 #11J71 #11J83 #28A78 #42A16 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2504.03575

Abstract

Let \((a_n)_{n ∈ ℕ}\) be a lacunary sequence of integers satisfying the Hadamard gap condition. For any fixed dimension d ≥ 1, we establish asymptotic upper bounds for the maximal gap in the set of dilates \(\\boldsymbolα a_n \_{n ≤ N}\) modulo 1 as N → ∞, for Lebesgue--almost all dilation vectors \boldsymbolα ∈ [0,1]^d. More precisely, we prove that for any lacunary \((a_n)_{n ∈ ℕ}\) and Lebesgue--almost all \boldsymbolα, every convex set in [0,1]^d of volume at least (log N)^2+ε/N must contain an element of the set \(\\boldsymbolα a_n \_{n ≤ N}\) 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.

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