Forbidden sparse intersections

Abstract

Let $n$ be a positive integer, let $0<p⩽p^{\prime }⩽\frac{1}{2}$, and let $\ell ⩽pn$ be a nonnegative integer. We prove that if $\mathcal{F},\mathcal{G}\subseteq \{0,1{\}}^n$ are two families whose cross intersections forbid $\ell$ -- that is, they satisfy $|A\cap B|\ne \ell$ for every $A\in \mathcal{F}$ and every $B\in \mathcal{G}$ -- then, setting $t:=\min \{\ell ,pn-\ell \}$, we have the subgaussian bound \[ μ_p(\mathcal{F})\, μ_{p'}(\mathcal{G})\leqslant 2\exp\Big( - \frac{t^2}{58^2\,pn}\Big), \] where $\hat{I}{¼}_p$ and $\hat{I}{¼}_{p^{\prime }}$ denote the $p$-biased and $p^{\prime }$-biased measures on $\{0,1{\}}^n$ respectively.