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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]

Ernie Croot and Olof Sisask introduced a sampling technique for approximate translation invariance in additive combinatorics. Their Croot-Sisask almost-periodicity theorem controls how much a convolution changes under many translations.^[ref]

The useful shift is from exact algebraic symmetry to small error in a function norm. Random samples approximate averages; comparing repeated approximations exposes almost-periods. The original paper carefully specifies counting convolution and counting $L^p$ norms, which determine the constants and scale of the assertion.^[ref]

Olof Sisask co-developed the Croot-Sisask almost-periodicity theorem with Ernie Croot, introducing a probabilistic method for finding almost-periods of convolutions in additive combinatorics.^[ref]

Their proof approximates a convolution by sampling and then looks for translations preserving the approximation. Norm estimates turn that probabilistic approximation into many approximate symmetries. The method provides a way to detect additive structure without requiring exact periodicity.^[ref]