Rado覆盖问题的指数级改进

中文摘要 AI 辅助

设$B^d$表示$d$维单位半径的欧几里得球。最大的常数$f(B^d) \in [0,1]$满足：$\mathbb{R}^d$中任何有限个单位球的集合$\mathcal{C}$都存在一个不相交的子集$\mathcal{S}$，其占据$\mathcal{C}$体积的至少$f(B^d)$比例。这个问题最早由T. Radó在1928年针对平面上的轴平行正方形提出；作者受到Vitali在实分析中经典覆盖引理的启发。欧几里得球的情形最早由R. Rado在1949年考虑。直到去年，关于单位球的$f(B^d)$的最佳已知估计相差甚远：\\[(1+\epsilon_d) 3^{-d} \leq f(B^d) \leq 2^{-d},\\]其中$0<\epsilon_d=o_{d\rightarrow \infty}(1)$。最近，本注记的作者观察到，上界的指数级改进可由Kabatiansky--Levenshtein球面码界得出，而下界则由C. Xie和G. Ge（见arXiv:2608.09744）改进了一个线性因子。当前对于大$d$的最佳估计为：\\[c \cdot d \cdot 3^{-d} \leq f(B^d) \leq 2.447^{-d},\\]其中$c>0$是绝对常数。这里我们提供了近80年来下界的首次指数级改进，将差距缩小到：\\[2.910^{-d} \leq f(B^d) \leq 2.447^{-d}.\\]我们的方法是构造性的，并产生一个多项式时间算法来找到实现该估计的不相交子集。此外，同样的技术对所有满足一致凸性假设的对称凸体（例如所有$p\in (1,\infty)$的$\ell^p$-球）给出类似的指数级改进下界。

英文摘要

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 $p\in (1,\infty)$.

发表机构

• Istituto Nazionale di Alta Matematica “F. Severi”（高等数学国家研究所“F. Severi”）
• Scuola Normale Superiore（比萨高等师范学院）
• Algoresearch L.L.C.（Algoreresearch有限责任公司）
• Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）
• Research Institute of the University of Bucharest（布加勒斯特大学研究院）

机构由 AI 辅助整理，请以论文原文为准。