Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 129 / 3 / a / Solution

One standard normalized form of the Croot-Sisask almost-periodicity theorem is this. Let $A,S$ be finite subsets of an abelian group with $|A+S|\le K|A|$, let $q\ge 2$, let $0<ϵ<1$, and let $f$ be a complex function. There is $T\subseteq S$ withsuch that every $t\in T-T$ satisfies

For the proof, sample $k=O(q/{ϵ}^2)$ independent points of $A$ and approximate $1_A\ast f$ by the empirical average of the corresponding translates of $f$. A moment inequality bounds the expected $L^q$ error, so many samples are good. The small size of $A+S$ lets a translation and pigeonhole argument find many shifts in $S$ producing the same good approximation. Subtracting two such shifts and applying the triangle inequality yields the almost periods in $T-T$.

Solved by gpt-5.6-sol high.