Finite-field convolution almost-periodicity theorem

Let $G={\mathbb{F}}_p^n$, $m\ge 1$ be an integer, and $\epsilon >0$. Use counting convolution,and the counting Lp norm $\Vert F{\Vert }_{{\ell }^{2m}}=({\sum }_x|F(x){|}^{2m}{)}^{1/(2m)}$. There is a vector subspace $V\le G$ of codimension of a vector subspace $O(m{\epsilon }^{-2})$ such that every $t\in V$ satisfiesEquivalently, with normalized convolution on a finite group $f=F/|G|$, normalized Lp norms and subset density $\alpha =|A|/|G|$, the error is $\epsilon \alpha$.

For nonempty $A$, Parseval identity gives ${\sum }_{\chi }|\widehat{1_A}(\chi ){|}^2=\alpha$, so the Fourier coefficients on a finite abelian group of $f$ have total absolute sum $\alpha$. Sampling $O(m{\epsilon }^{-2})$ phase-adjusted characters, using the Marcinkiewicz–Zygmund inequality, approximates $f$ in normalized $L^{2m}$ within $\epsilon \alpha /2$. The common kernel of these characters has at most that codimension of a vector subspace. The approximation is unchanged by every shift in that kernel; the triangle inequality gives the stated error for $f$. If $A$ is empty, take $V=G$.