AI 中文总结
本文针对$\omega(\ell)\geq2$的模复合数Oddtown问题，改进了其最大集合族大小的上界，依赖相关引理与猜想完成证明。
AI 中文摘要
设$f_{\ell}(n)$为满足以下条件的最大集合族$\mathcal{A}\subseteq2^{[n]}$的大小：无成员的大小被$\ell$整除，且任意两个不同成员的交集大小被$\ell$整除；记$\omega(\ell)$为$\ell$的不同素因子个数。对于任意素数幂$\ell$，经典结论为$f_{\ell}(n)=n$。当$\omega(\ell)\geq2$时，Bukh、Chao和Zheng近期证明：存在$C_{\ell}>0$，使得$\omega(\ell)n-O_{\ell}\left(n^{\frac{\omega(\ell)-2}{\omega(\ell)-1}}(\log n)^{C_{\ell}}\right)\leq f_{\ell}(n)\leq\omega(\ell)n-2\omega(\ell)\log n+11$。当$\ell$至少有两个不同奇素因子时，他们还利用傅里叶分析将上界改进为存在$\varepsilon_{\ell}>0$，当$n$关于$\ell$足够大时，$f_{\ell}(n)\leq\omega(\ell)n-(2\omega(\ell)+\varepsilon_{\ell})\log n$。本文对每个固定的$\omega(\ell)\geq2$的$\ell$，证明当$n$足够大时，$f_{\ell}(n)\leq\omega(\ell)n-\Omega_{\ell}(\log n\log\log n)$。该上界依赖于Bhowmick、Dvir和Lovett的子矩阵引理，而该引理基于Gowers、Green、Manners和Tao近期证明的有界挠多项式Freiman–Ruzsa猜想。
英文摘要
Let $f_{\ell}(n)$ be the largest size of a family $\mathcal{A}\subseteq2^{[n]}$ such that no member has size divisible by $\ell$, while the intersection of every two distinct members has size divisible by $\ell$, and let $ω(\ell)$ denote the number of distinct prime divisors of $\ell$. For any prime power $\ell$, the classical answer is $f_{\ell}(n)=n$. When $ω(\ell)\geq 2$, Bukh, Chao, and Zheng recently proved $ω(\ell)n-O_{\ell}\left(n^{\frac{ω(\ell)-2}{ω(\ell)-1}}(\log n)^{C_{\ell}}\right)\leq f_{\ell}(n)\leqω(\ell)n-2ω(\ell)\log n+11$ for some $C_{\ell}>0$. When $\ell$ has at least two distinct odd prime divisors, they further used Fourier analysis to improve the upper bound to $f_{\ell}(n)\leqω(\ell)n-(2ω(\ell)+\varepsilon_{\ell})\log n$ for some $\varepsilon_{\ell}>0$, provided that $n$ is sufficiently large in terms of $\ell$ . For every fixed $\ell$ with $ω(\ell)\geq2$, we prove \[ f_{\ell}(n)\leqω(\ell)n-Ω_{\ell}(\log n\log\log n) \] for large $n$. The upper bound relies on a submatrix lemma of Bhowmick, Dvir, and Lovett, which is based on the bounded-torsion polynomial Freiman--Ruzsa conjecture recently proved by Gowers, Green, Manners, and Tao.
