Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 129 / 3 / i / Solution

One normalized form of the Croot-Sisask almost-periodicity theorem is as follows. If finite sets $A,S$ in a group satisfy $|A+S|\le K|A|$, $p\ge 2$, and $0<ϵ<1$, then for every function $f$ there is $T\subseteq S$ withsuch that