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

2026-08-10

AI Summary

This study addresses the problem of selecting a pairwise disjoint subcollection from a finite family of congruent balls in Euclidean space such that its total volume constitutes at least a fixed proportion of the volume of the entire union. Building upon the classical Rado covering problem, the work introduces for the first time a weighted hard-core model, combining combinatorial arguments with weighted geometric estimates that capture the overlap structure of balls. This approach yields a substantial improvement over classical Vitali-type lower bounds. The main contributions include establishing, for any dimension \(d\), the bound \(f(B^d) \geq \frac{2}{3^d + 2^d}\), and demonstrating that for sufficiently large \(d\), the resulting asymptotic lower bound improves upon the classical \(3^{-d}\) by approximately a factor of \(d\).

Abstract

Let $B^d$ denote the Euclidean unit ball in $\mathbb{R}^d$ and $f(B^d)$ denote the largest constant $c$ such that every finite collection of congruent Euclidean balls 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 $f(B^d)\geq3^{-d}$. In this paper, we establish two improvements. First, by a purely combinatorial argument, we prove that $$ f(B^d)\geq \frac{2}{3^d + 2^d} $$ for every integer $d \geq1$. 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 $d$, $$ f(B^d)\geq \left( \log\frac{3}{1+\sqrt3} -O\left(\frac{\log d}{d}\right) \right)d\,3^{-d}. $$ Thus, the classical lower bound is improved by a factor of order $d$.

Problem

• covering problem
• Euclidean balls
• Vitali covering lemma
• disjoint subcollection
• volume ratio

Innovation

• weighted hard-core model
• Rado's covering problem
• Vitali covering lemma
• Euclidean balls
• combinatorial geometry