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

For the counting-measure Lp norm, defineThis is the set of Lp almost periods of $f$ with exponent $p=2m$ and error $X$.

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.

Let $T=1_A\ast 1_A$ and chooseApply part b with its error parameter replaced by a sufficiently small absolute multiple of $\epsilon$. This produces a vector subspace $V$ of codimensionwith the evident harmless modification when $\alpha =1$.

For $t\in V$, $x\in {\mathbb{F}}_p^n$, and $q=2m$, Hölder's inequality givesSince ${\alpha }^{-1/q}\le e^{1/2}$ by the choice of $m$, the bound from part b is at mostafter absorbing the absolute factor into the chosen error parameter. This is the required uniform estimate for $1_A\ast 1_A\ast 1_A$.

The sum over all $x$ of the threefold-convolution representation function is $|A{|}^3$. Some $x_0$ therefore satisfiesUse part c with $\epsilon =\alpha /2$. There is a vector subspace $V$ of codimension $O({\alpha }^{-2}\log (1/\alpha ))$ such that, for every $t\in V$,Positivity means that $x_0+t$ has a representation as a sum of three elements of $A$. Thuswhich is the desired translate of a low-codimension subspace.