Finite-field convolution almost-periodicity theorem

If $A\subseteq {\mathbb{F}}_p^n$, $m\ge 1$, and $\epsilon >0$, then the $L^{2m}$ almost periods of $1_A\ast 1_A$ with error $\epsilon |A|p^{n/(2m)}$ contain a vector subspace of codimension $O(m{\epsilon }^{-2})$. One proof samples the Fourier expansion of the convolution using the Marcinkiewicz–Zygmund inequality; the sampled characters have a common kernel of the required codimension.