The higher-dimensional Shepp problem: an exact criterion for random ball coverings of tori
Abstract
We solve the Euclidean-ball case of the higher-dimensional Shepp covering problem. More precisely, we give an exact criterion for full limsup coverage of the d-dimensional torus, d≥ 2, by independently centered Euclidean balls with an arbitrary decreasing sequence of radii. Let X1,X2,… be independent Haar-uniform points, let r1≥ r2≥⋯↓ 0, and put un(z)=m(B(0,rn)∩ B(z,rn)) and H(z)=∑n≥ 1un(z). Then every point belongs to infinitely many of the balls B(Xn,rn) almost surely if and only if ∫𝕋dexp(H) dm=∞. In dimension one this condition is equivalent to Shepp's criterion. No regular-variation or comparable-scale assumption is imposed on the radii. The main difficulty is shared noise: after spatial decomposition, the same Poisson input acts on many uncovered cells, so their descendants are not conditionally independent. We overcome this by establishing an extinction bound for monotone population recursions driven by positively associated innovations. Together with Poissonization and spatial localization this proves sufficiency, while a second-moment estimate and the zero-one law prove necessity.
Related
- The Bernstein problem in higher dimensional Heisenberg groups
- Minimal Covering Bodies and a Minkowski-Type Criterion for Lattice Coverings
- An Island-Based Parallel Biased Random-Key Genetic Algorithm for the Three-Dimensional Trailer Loading Problem
- Intermediate coverings, random images, and projections
- Intractable enumeration problems are like Russian nesting dolls: structural properties of monomer-dimer coverings on two-dimensional quadratic lattices
BibTeX
@misc{zhou2026higherdimensional,
title = {{The higher-dimensional Shepp problem: an exact criterion for random ball coverings of tori}},
author = {Yechi Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02156},
eprint = {2609.02156},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02156},
}
RIS
TY - UNPB
TI - The higher-dimensional Shepp problem: an exact criterion for random ball coverings of tori
AU - Yechi Zhou
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02156
UR - https://arxiv.org/abs/2609.02156
AB - We solve the Euclidean-ball case of the higher-dimensional Shepp covering problem. More precisely, we give an exact criterion for full limsup coverage of the d-dimensional torus, d≥ 2, by independently centered Euclidean balls with an arbitrary decreasing sequence of radii. Let X_1,X_2,… be independent Haar-uniform points, let r_1≥ r_2≥⋯↓ 0, and put u_n(z)=m(B(0,r_n)∩ B(z,r_n)) and H(z)=∑_n≥ 1u_n(z). Then every point belongs to infinitely many of the balls B(X_n,r_n) almost surely if and only if ∫_𝕋^dexp(H) dm=∞. In dimension one this condition is equivalent to Shepp's criterion. No regular-variation or comparable-scale assumption is imposed on the radii. The main difficulty is shared noise: after spatial decomposition, the same Poisson input acts on many uncovered cells, so their descendants are not conditionally independent. We overcome this by establishing an extinction bound for monotone population recursions driven by positively associated innovations. Together with Poissonization and spatial localization this proves sufficiency, while a second-moment estimate and the zero-one law prove necessity.
ER -