Croot-Sisask almost-periodicity theorem

In an abelian group $G$, let $A,S$ be nonempty finite sets with $|A+S|\le K|A|$. For $q\ge 2$, $0<\epsilon <1$ and $f\in {\ell }^q(G)$, there is $T\subseteq S$ withsuch that every $t\in T-T$ is an Lp almost period of the counting convolution $1_A\ast f$:Equivalently, average over $A$ by ${\mu }_A=1_A/|A|$ and replace the right side by $\epsilon \Vert f{\Vert }_{{\ell }^q}$. This fixes the same counting convention in the convolution and norm.^[ref]

The proof samples translates of $f$. A moment bound makes most samples approximate the average over $A$; small growth of $A+S$ forces many shifts in $S$ to share a good sample. Differences between these shifts give the almost periods. The original method also works in noncommutative groups, where the order of products and side of translation must be specified.^[ref]