Exponential improvements in Rado's covering problem

Abstract

Let $B^d$ denote the $d$-dimensional Euclidean ball of unit radius. What is the largest constant $f(B^d) \in [0,1]$ with the property that every finite collection $\mathcal{C}$ of unit balls in $\mathbb{R}^d$ admits a disjoint sub-collection $\mathcal{S}$ occupying at least a fraction $f(B^d)$ of the volume of $\mathcal{C}$? This problem was first raised by T. Radó in 1928, for axis-parallel squares in the plane; the author was motivated by a classical covering lemma in real analysis due to Vitali. The case of Euclidean balls was first considered by R. Rado in 1949. Until last year the best known estimates on $f(B^d)$ for unit balls where very far apart: \[ (1+ε_d) 3^{-d} \leq f(B^d) \leq 2^{-d}, \] where $0<ε_d=o_{d\rightarrow \infty}(1)$. Recently, the authors of this note observed that an exponential improvement on the upper bound follows from the Kabatiansky--Levenshtein spherical code bound, while the lower bound was improved by a linear factor by C.~Xie and G.~Ge (see arxiv:2608.09744). The current best estimates for large $d$ are \[ c \cdot d \cdot 3^{-d} \leq f(B^d) \leq 2.447^{-d}, \] where $c>0$ is an absolute constant. Here we offer the first exponential improvement of the lower bound in almost 80 years, which narrows the gap to: \[ 2.910^{-d} \leq f(B^d) \leq 2.447^{-d}. \] Our method is constructive and yields a polynomial time algorithm for finding a disjoint sub-collection realizing the estimate. Moreover the same technique gives similar exponentially improved lower bounds for all symmetric convex bodies satisfying a uniform convexity assumption, e.g., $\ell^p$-balls for all $p\in (1,\infty)$.