Finite-field convolution almost-periodicity theorem
= Finite-field convolution almost-periodicity theorem
If $A\subseteq\mathbb F_p^n$, $m\geq1$, and $\varepsilon>0$, then the $L^{2m}$ almost periods of $1_A*1_A$ with error $\varepsilon|A|p^{n/(2m)}$ contain a <vector subspace> of codimension $O(m\varepsilon^{-2})$. 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.