Forbidden sparse intersections

Let $n$ be a positive integer, let $0<p⩽p^{\mathrm{\prime }}⩽\frac{1}{2}$, and let $\mathrm{\ell }⩽pn$ be a nonnegative integer. We prove that if $\mathcal{F},\mathcal{G}\subseteq \{0,1{\}}^n$ are two families whose cross intersections forbid $\mathrm{\ell }$ – that is, they satisfy $\mathrm{\mid }A\cap B\mathrm{\mid }\mathrm{\ne }\mathrm{\ell }$ for every $A\in \mathcal{F}$ and every $B\in \mathcal{G}$ – then, setting $t:=\min \{\mathrm{\ell },pn-\mathrm{\ell }\}$, we have the subgaussian bound $${\mu }_p(\mathcal{F}){\mu }_{p^{\mathrm{\prime }}}(\mathcal{G})⩽2\exp (-\frac{t^2}{{58}^{2}pn}),$$ where ${\mu }_p$ and ${\mu }_{p^{\mathrm{\prime }}}$ denote the $p$-biased and $p^{\mathrm{\prime }}$-biased measures on $\{0,1{\}}^n$ respectively.