Covering large-dimensional Euclidean spaces by random translates of a given convex body

TL;DR

The paper studies the high-dimensional problem of covering $\mathbb{R}^n$ by translates of a convex body, focusing on Euclidean balls. It shows the existence of ball coverings with density $\vartheta(P)\le (\tfrac{1}{2}+o(1))\,n \ln n$ while keeping the maximum multiplicity $\mu(P)\le (\xi+o(1))\,n \ln n$ with $\xi=1.79556...$, improving prior bounds; simultaneously, it proves that the standard Poisson/torus random construction cannot beat the $(\tfrac{1}{2}+o(1))\,n \ln n$ barrier for any convex body, and identifies the cube as a worst-case example for this method, with packing density $$(1+o(1))\,n \ln n.$$ The proofs combine Poisson-process constructions on packing tori, concentration arguments, and isotropic-convex-body tools (including the isotropic constant bound). These results sharpen the density–multiplicity trade-off in high dimensions and delineate the power and limits of a widely used random-construction paradigm for coverings.

Abstract

Determining the minimum density of a covering of $\mathbb{R}^{n}$ by Euclidean unit balls as $n\to\infty$ is a major open problem, with the best known results being the lower bound of $\left(\mathrm{e}^{-3/2}+o(1)\right)n$ by Coxeter, Few and Rogers [Mathematika 6, 1959] and the upper bound of $\left(1/2+o(1) \right)n \ln n$ by Dumer [Discrete Comput. Geom. 38, 2007]. We prove that there are ball coverings of $\mathbb{R}^n$ attaining the asymptotically best known density $\left(1/2+o(1) \right)n \ln n$ such that, additionally, every point of $\mathbb{R}^n$ is covered at most $\left(1.79556... + o(1)\right) n \ln n$ times. This strengthens the result of Erdős and Rogers [Acta Arith. 7, 1961/62] who had the maximum multiplicity at most $\left(\mathrm{e} + o(1)\right) n \ln n$. On the other hand, we show that the method that was used for the best known ball coverings (when one takes a random subset of centres in a fundamental domain of a suitable lattice in $\mathbb{R}^n$ and extends this periodically) fails to work if the density is less than $(1/2+o(1))n\ln n$; in fact, this result remains true if we replace the ball by any convex body $K$. Also, we observe that a ``worst'' convex body $K$ here is a cube, for which the packing density coming from random constructions is only $(1+o(1))n\ln n$.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Lemma 2.1
• Lemma 2.2
• proof : Proof of Lemma \ref{['LEMMA:volume-intersection-sphere']}
• proof : Proof of Theorem \ref{['THM:poisson-process-upper-bound']}
• Claim 2.3