Moderate-doubling sets in ${\mathbb{F}}_2^n$ intersect subspaces

We show that any set $A$ in ${\mathbb{F}}_2^n$ with $\mathrm{\mid }A+A\mathrm{\mid }\le \mathrm{\mid }A{\mathrm{\mid }}^{2-\eta }$ must intersect a subspace of dimension $O_{\eta }(\log \mathrm{\mid }A\mathrm{\mid })$ in at least $\mathrm{\mid }A{\mathrm{\mid }}^{\eta -o(1)}$ elements.