= Croot-Sisask almost-periodicity theorem
{c}
In an <abelian group> $G$, let $A,S$ be nonempty finite <sets> with $|A+S|\leq K|A|$. For $q\geq2$, $0<\varepsilon<1$ and $f\in\ell^q(G)$, there is $T\subseteq S$ with
$$
|T|\geq(2K)^{-O(q/\varepsilon^2)}|S|
$$
such that every $t\in T-T$ is an <Lp almost period> of the counting <convolution> $1_A*f$:
$$
\|\tau_t(1_A*f)-1_A*f\|_{\ell^q}\leq\varepsilon|A|\|f\|_{\ell^q},\qquad \tau_tF(x)=F(x+t).
$$
Equivalently, average over $A$ by $\mu_A=1_A/|A|$ and replace the right side by $\varepsilon\|f\|_{\ell^q}$. This fixes the same counting convention in the <convolution> and <norm>.<Notes on the Croot-Sisask lemma by Kevin Henriot>[Theorem 2, page 2, specialized to an abelian group; section 2 conventions]{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.<A probabilistic technique for finding almost-periods of convolutions by Ernie Croot and Olof Sisask>[Sections 1.1 and 3]{ref}
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