A superlogarithmic saving for Oddtown modulo composite numbers

Abstract

Let $f_{\ell }(n)$ be the largest size of a family $\mathcal{A}\subseteq 2^{[n]}$ such that no member has size divisible by $\ell$, while the intersection of every two distinct members has size divisible by $\ell$, and let $\omega (\ell )$ denote the number of distinct prime divisors of $\ell$. For any prime power $\ell$, the classical answer is $f_{\ell }(n)=n$. When $\omega (\ell )\ge 2$, Bukh, Chao, and Zheng recently proved $\omega (\ell )n-O_{\ell }\left(n^{\frac{\omega (\ell )-2}{\omega (\ell )-1}}(\log n{)}^{C_{\ell }}\right)\le f_{\ell }(n)\le \omega (\ell )n-2\omega (\ell )\log n+11$ 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 $f_{\ell }(n)\le \omega (\ell )n-(2\omega (\ell )+{\epsilon }_{\ell })\log n$ for some ${\epsilon }_{\ell }>0$, provided that $n$ is sufficiently large in terms of $\ell$ . For every fixed $\ell$ with $\omega (\ell )\ge 2$, 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.