On the weighted hard-core model and Rado's covering problem for congruent Euclidean balls

Abstract

Let $K$ be a symmetric convex body in ${\mathbb{R}}^d$ and let $f(K)$ denote the largest constant $c$ such that every finite collection of translates of $K$ contains a pairwise disjoint subcollection whose total volume is at least $c$ times the volume of the union of the original collection. The classical Vitali covering lemma gives $f(K)\ge 3^{-d}$. In this paper, we establish two improvements. First, by a purely combinatorial argument, we prove that $$f(K)\ge \frac{2}{3^d+2^d}$$ for every symmetric convex body $K$. This improves the Vitali bound by a factor tending to $2$ as $d$ tends to infinity. Second, using a weighted hard-core model together with a weighted geometric estimate for intersections of Euclidean balls, we show that, for all sufficiently large $d$, $$f(B^d)\ge \left(\log \frac{3}{1+\sqrt{3}}-O\left(\frac{\log d}{d}\right)\right)d{3}^{-d},$$ where $B^d$ is the unit Euclidean ball in ${\mathbb{R}}^d$. Thus, the classical lower bound is improved by a factor of order $d$.

arXiv comment17 pages; any comments are welcome