发表机构
Fudan University（复旦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究解决了高维Shepp覆盖问题的欧氏球情形，给出了d≥2维环面随机球完全上极限覆盖的精确判据，克服共享噪声困难并证明了判据的充分性与必要性。
AI 中文摘要
我们解决了高维Shepp覆盖问题的欧氏球情形。更准确地说，对于任意递减半径序列，我们给出了d维环面（d≥2）被以独立均匀分布点为中心的欧氏球完全上极限覆盖的精确判据。设X₁,X₂,…为独立Haar均匀点，r₁≥r₂≥…↓0，令uₙ(z)=m(B(0,rₙ)∩B(z,rₙ))，H(z)=∑ₙ≥1uₙ(z)。则几乎必然每个点属于无穷多个球B(Xₙ,rₙ)当且仅当∫_{𝕋ᵈ}exp(H)dm=∞。在一维情形，该条件等价于Shepp判据。我们未对半径施加正则变化或可比尺度假设。主要困难是共享噪声：空间分解后，同一泊松输入作用于多个未覆盖单元，故它们的后代不条件独立。我们通过建立由正相关创新驱动的单调种群递归的灭绝界克服了这一困难，结合泊松化与空间定位证明了充分性，而二阶矩估计与零一律证明了必要性。
英文摘要
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.
Comments23 pages. Accompanying Lean 4 formalization: https://github.com/zyc111-234/shepp-formalization
