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, $d\ge 2$, by independently centered Euclidean balls with an arbitrary decreasing sequence of radii. Let $X_1,X_2,\ldots$ be independent Haar-uniform points, let $r_1\ge r_2\ge\cdots\downarrow 0$, and put $u_n(z)=m(B(0,r_n)\cap B(z,r_n))$ and $H(z)=\sum_{n\ge 1}u_n(z)$. Then every point belongs to infinitely many of the balls $B(X_n,r_n)$ almost surely if and only if $\int_{\mathbb{T}^d}\exp(H) dm=\infty$. 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.