Rado's covering problem for cubes and balls: a semi-survey

TL;DR

This paper investigates Rado's covering problem for high-dimensional cubes and balls, providing bounds on the largest fraction of area that can be covered by disjoint subcollections, with a focus on asymptotic behavior as dimension increases.

Contribution

The paper derives new bounds for the covering constants of high-dimensional cubes and balls, improving understanding of Rado's problem in the asymptotic regime.

Findings

For cubes, the bounds are ^{-1} rac{2^{-d}}{d \u2217 \u03bb d} \u2264 F(Q^d) ; upper bound is 2^{-d}.

For balls, bounds are (1+\u03b5_d)3^{-d}   2.447^{-d}, with     exponentially fast.

Upper bounds for high-dimensional balls are obtained using the Kabatiansky-Levenshtein sphere packing bound.

Abstract

What is the largest constant $c\in [0,1]$ with the property that every finite collection $\mathcal{C}$ of axis-parallel squares in the plane admits a disjoint sub-collection $\mathcal{S}$ occupying at least a fraction $c$ of the area covered by $\mathcal{C}$? This problem was first raised by T.~Rad\'o in 1928, who was motivated by a classical covering lemma in real analysis due to Vitali. R.~Rado later generalized the problem from axis-parallel squares in the plane to homothetic copies of any given convex body $K$ in $\mathbb{R}^d$, where now we are looking for an optimal constant $F(K)$. Our utmost interest is for cubes and balls in the high-dimensional regime $d\rightarrow \infty$. The estimates that we currently have for cubes are much more precise than those for balls: namely if $Q^d$ is a $d$-dimensional cube, then \[ (e^{-1}+o(1))\frac{2^{-d}}{d \log{d}} \leq F(Q^d)\leq 2^{-d}, \] while denoting $B^d$ a $d$-dimensional Euclidean ball, then \[ (1+\epsilon_d)3^{-d}\leq F(B^d)\leq 2.447^{-d}, \] where $\epsilon_d>0$ vanishes exponentially fast as $d\rightarrow \infty$. The latter upper bound is obtained here by using the Kabatiansky--Levenshtein bound for the sphere packing problem.