AI 中文总结
研究覆盖\(\mathbb{R}^n\)中特定有限集所需直径为\(1\)的欧几里得球最大数量\(g_n\)，通过率失真理论证明指数增长率为\(\alpha_0\)，给出两点情形下界及三点构造，表明\(\alpha_0\)不由有限支撑分布达到。
AI 中文摘要
设\(g_n\)为覆盖\(\mathbb{R}^n\)中直径为\(1\)的集合所需直径为\(1\)的欧几里得球的最大数量。我们研究了在所有坐标置换下不变的有限集的这个问题。证明了此对称问题中的指数增长率可精确表征为有限字母平方误差率失真上确界\(\alpha_0\)。在两点情形下，即对于布尔立方体的子集，给出明确下界\(g_n\geq(1.160235457\ldots - o(1))^n\)，改进了先前最佳界\((2/\sqrt{3}-o(1))^n\)。利用菲克斯对率失真问题的高斯表征，给出指数基大于\(1.160497831\)的数值三点构造。最后表明\(\alpha_0\)不由任何有限支撑分布达到。
英文摘要
Let $g_n$ be the largest number of Euclidean balls of diameter $1$ which may be needed to cover a set of diameter $1$ in $\mathbb{R}^n$. We study this problem for finite sets invariant under all coordinate permutations. We prove that the exponential growth rate in this symmetric problem can be characterized exactly as a finite-alphabet squared-error rate-distortion supremum $α_0$. Specialized to the two-point case, i.e., for subsets of Boolean cubes, this gives the explicit lower bound \[g_n\ge (1.160235457\ldots-o(1))^n,\] improving the previous best bound $(2/\sqrt3-o(1))^n$. Using Fix's Gaussian characterization of the rate-distortion problem, we give a numerical three-point construction with exponent base greater than $1.160497831$. Finally, we show that $α_0$ is not attained by any finitely supported distribution.
Comments15 pages
