The Oddtown problem modulo a composite number

Abstract

A family of subsets $\mathcal{A}$ of an $n$-element set is called an $\ell$-Oddtown if the sizes of all sets are not divisible by $\ell$, but the sizes of pairwise intersections are divisible by $\ell$. Berlekamp and Graver showed that when is a $\ell$ is a prime, the maximum size of an $\ell$-Oddtown is $n$. For composite moduli with $\omega$ distinct prime factors, the argument of Szegedy gives an upper bound of $\omega n-\omega\log_2 n$ on the size of an $\ell$-Oddtown. We improve this to $\omega n-(2\omega +\varepsilon)\log_2 n$ for most $\ell$ and $n$ using a combination of linear algebraic and Fourier-analytic arguments.