正则3-相交族的定量界
Quantitative bounds for regular $3$-wise intersecting families
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中文总结 AI 辅助
本文针对正则递增3-相交族，用初等工具给出定量界的简短证明，还通过傅里叶分析得到较弱估计，改进了相关研究的定量结果。
中文摘要 AI 辅助
Frankston、Kahn和Narayanan利用Friedgut的 junta 定理证明，[n]的子集构成的每个正则递增3-相交族的基数为o(2ⁿ)。本文用布尔函数分析和熵的初等工具给出一个简短的定量证明：若𝒜⊆𝒫ₙ是非空正则递增3-相交族，则log(2ⁿ/|𝒜|)≥(n/2)(|𝒜|/(2ⁿ-|𝒜|))²，故|𝒜|≤2ⁿ√(W(n)/n)，其中W是满足W(x)e^W(x)=x（x≥0）的主Lambert函数；还给出纯傅里叶分析方法得到的较弱估计|𝒜|≤2ⁿ/(1+n^(1/3))。
英文摘要
Frankston, Kahn and Narayanan proved that every regular increasing $3$-wise intersecting family of subsets of $[n]$ has cardinality $o(2^n)$ using Friedgut's junta theorem. We give a short quantitative proof using elementary tools from the analysis of Boolean functions and entropy. More precisely, if $\mathcal{A}\subseteq\mathcal{P}_n$ is a nonempty $3$-wise intersecting family that is both regular and increasing, then $$ \log\frac{2^n}{|\mathcal{A}|}\ge \frac{n}{2}\left(\frac{|\mathcal{A}|}{2^n-|\mathcal{A}|}\right)^2, $$ and consequently $|\mathcal{A}|\le 2^n\sqrt{W(n)/n}$, where $W$ is the principal Lambert function defined by $W(x)e^{W(x)}=x$ for $x\ge0$. We also give a purely Fourier-analytic proof of the weaker estimate $$ |\mathcal{A}|\le \frac{2^n}{1+n^{1/3}}. $$
