Forbidden sparse intersections

Let $n$ be a positive integer, let $0<p\leqslant p'\leqslant \frac{1}{2}$, and let $\ell \leqslant pn$ 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 $|A\cap B|\neq \ell$ for every $A\in\mathcal{F}$ and every $B\in\mathcal{G}$ -- then, setting $t:=\min\{\ell,pn-\ell\}$, we have the subgaussian bound $$\mu_p(\mathcal{F})\, \mu_{p'}(\mathcal{G})\leqslant 2\exp\Big( - \frac{t^2}{58^2\,pn}\Big),$$ where $\mu_p$ and $\mu_{p'}$ denote the $p$-biased and $p'$-biased measures on $\{0,1\}^n$ respectively.