$\mathbb F_2^n$中仿射子空间统计量的渐近紧界
Asymptotically sharp bounds for affine subspace statistics in $\mathbb F_2^n$
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中文总结 AI 辅助
针对$\mathbb F_2^n$中的仿射子空间统计问题，本文推导了$n$趋于无穷时概率最大值$\lambda^*(d,s)$的两类紧界，给出了$s=j2^k$情形的匹配上界并精确求解了$s=1$时的最优值。
中文摘要 AI 辅助
给定子集$A \subseteq \mathbb F_2^n$，我们可以考虑$A$与均匀随机$d$-平坦集$F$的交集大小的分布。受边统计问题和超立方体统计问题的启发，仿射子空间统计问题研究的是：对于任意固定的$s\in\{1,\dots,2^d\}$，在所有$A \subseteq \mathbb F_2^n$中，均匀随机$d$-平坦集$F$满足$|F\cap A|=s$的概率$\mathbb{P}[|F\cap A|=s]$的最大值。我们用$\lambda^*(d,s)$表示$n$趋于无穷时该最大值的极限。在本短文中，我们证明了两种不同情形下$\lambda^*(d,s)$的紧界。当$s=j2^k$（其中$j$为正奇数）时，目前已知的达到$\lambda^*(d,s)\ge 1-2^{-k}$的最优下界构造是将$A$取为$\mathbb F_2^n$中$j$个平行的$(n-d+k)$-平坦集的并集；我们的主要结果是给出了带有$O(2^{-3k/2})$加性误差项的匹配上界。我们还研究了$s=1$的情形，精确确定了$\lambda^*(d,1)$的值，证明了每个点以$2^{-d}$概率被纳入的随机构造是最优的。
英文摘要
Given a subset $A \subseteq \mathbb F_2^n$, we can consider the distribution of the intersection size of $A$ with a uniformly random $d$-flat $F$. Motivated by the edge statistics problem and the hypercube statistics problem, the affine subspace statistics problem concerns the maximum of $\mathbb{P}[|F\cap A|=s]$ among $A \subseteq \mathbb F_2^n$ for any fixed $s\in\{1,\dots,2^d\}$ over a uniformly random $d$-flat $F$. We use $λ^*(d,s)$ to denote the limit of the maximum when $n$ goes to infinity. In this note, we prove tight bounds for $λ^*(d,s)$ in two different regimes. For $s=j2^k$ where $j$ is a positive odd integer, the best known lower bound construction achieving $λ^*(d,s)\ge 1-2^{-k}$ is due to taking $A$ as the union of $j$ parallel $(n-d+k)$-flats in $\mathbb F_2^n$. Our main result is a matching upper bound with an additive error term of $O(2^{-3k/2})$. We also study the case $s=1$, where we determine $λ^*(d,1)$ exactly. We show that the random construction where each point is included with probability $2^{-d}$ is optimal.
