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Maximal volume of convex bodies avoiding random points and lacunary dilates
Abstract
For a positive real lacunary sequence and any probability measure μ with polynomial Fourier decay, we determine, for μ-almost every x, the asymptotic maximal volume of convex bodies avoiding the first N dilates of x modulo ℤd. Among homothets of a fixed convex body, the maximal volume is asymptotic to (log N)/N. When all convex bodies in the unit cube are allowed, it is asymptotic to d(log N)/N. For independent uniform points in a fixed convex body Ω⊂ℝd, the maximal volume of an empty convex body is asymptotic to dvol(Ω)(log N)/N almost surely. We also show, by constructing a one-dimensional counterexample, that lacunarity cannot be replaced by sparsity on sublacunary scales.
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BibTeX
@misc{peres2026maximal,
title = {{Maximal volume of convex bodies avoiding random points and lacunary dilates}},
author = {Yuval Peres and Bohan Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09790},
eprint = {2610.09790},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09790},
}
RIS
TY - UNPB
TI - Maximal volume of convex bodies avoiding random points and lacunary dilates
AU - Yuval Peres
AU - Bohan Yang
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.09790
UR - https://arxiv.org/abs/2610.09790
AB - For a positive real lacunary sequence and any probability measure μ with polynomial Fourier decay, we determine, for μ-almost every x, the asymptotic maximal volume of convex bodies avoiding the first N dilates of x modulo ℤ^d. Among homothets of a fixed convex body, the maximal volume is asymptotic to (log N)/N. When all convex bodies in the unit cube are allowed, it is asymptotic to d(log N)/N. For independent uniform points in a fixed convex body Ω⊂ℝ^d, the maximal volume of an empty convex body is asymptotic to dvol(Ω)(log N)/N almost surely. We also show, by constructing a one-dimensional counterexample, that lacunarity cannot be replaced by sparsity on sublacunary scales.
ER -