发表机构
University of Hawai‘i at Mānoa（夏威夷大学马诺阿分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定了Jaccard空间中度量维数的精确渐近常数为 $2n/\log_2 n$，通过将问题化为Erdős–Rényi称硬币问题，结合熵下界与显式检测族上界证明。
AI 中文摘要
设 $X$ 为满足 $|X|=n$ 的有限集，令 $\mathrm{Jac}(a,b)=|a\\,\triangle\\, b|/|a\cup b|$ 为幂集 $2^X$ 上的Jaccard距离。Lladser和Paradise最近证明了 $(2^X,\mathrm{Jac})$ 的度量维数为 $\Theta(n/\ln n)$，但常数未定；他们的界为 $(\ln 2)\\,n/\ln n\lesssim \beta(2^X,\mathrm{Jac})\lesssim 2\ln(2e)\\,n/\ln n$。我们确定了该常数：\\[ \beta(2^X,\mathrm{Jac})=\frac{2n}{\log_2 n}\\,(1+o(1))=(2\ln 2)\\,\frac{n}{\ln n}\\,(1+o(1)). \\] 证明将问题在固定基数的每个“切片”上等同于Erdős–Rényi弹簧秤称硬币问题（即“检测矩阵”问题）。下界是将Erdős–Rényi熵论证应用于中间切片；上界则来自Lindström以及Cantor和Mills的显式检测族，并辅以一个额外地标来揭示基数。
英文摘要
Let $X$ be a finite set with $|X|=n$ and let $\mathrm{Jac}(a,b)=|a\,\triangle\, b|/|a\cup b|$ be the Jaccard distance on the power set $2^X$. Lladser and Paradise recently proved that the metric dimension of $(2^X,\mathrm{Jac})$ is $Θ(n/\ln n)$, with the constant left open; their bounds are $(\ln 2)\,n/\ln n\lesssim β(2^X,\mathrm{Jac})\lesssim 2\ln(2e)\,n/\ln n$. We determine the constant: \[ β(2^X,\mathrm{Jac})=\frac{2n}{\log_2 n}\,(1+o(1))=(2\ln 2)\,\frac{n}{\ln n}\,(1+o(1)). \] The proof identifies the problem, on each ``slice'' of subsets of fixed cardinality, with the Erdős--Rényi coin-weighing problem for a spring scale (the problem of \emph{detecting matrices}). The lower bound is the Erdős--Rényi entropy argument applied to the middle slice; the upper bound follows from the explicit detecting families of Lindström and of Cantor and Mills, augmented by a single extra landmark that reveals cardinality.
