Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 129 / 3 / b / Solution

Put $N=p^n$, $\alpha =|A|/N$, and use normalized Fourier analysis on a finite abelian group. If $F=1_A\ast 1_A$ denotes unnormalized convolution and $h=N^{-1}F$, thenby the Parseval identity.

Sample $k=O(m{\epsilon }^{-2})$ characters independently, choosing $\gamma$ with probability $|\hat{h}(\gamma )|/\alpha$, and attach the phase of $\hat{h}(\gamma )$ to the sampled character. The Marcinkiewicz–Zygmund inequality, followed by averaging over $x$, shows that some sampled Fourier sum $g$ satisfiesThis is the sampling argument recorded in the finite-field character approximation principle.

Let $V$ be the intersection of the kernels of the sampled characters. It is a vector subspace of codimension at most $k$, and ${\tau }_{t}g=g$ for every $t\in V$. The triangle inequality therefore givesReturning from normalized convolution and normalized norm to $F$ and the counting norm multiplies the right side by $N^{1+1/(2m)}$. Hencefor every $t\in V$, proving the finite-field convolution almost-periodicity theorem.