Almost period of a function

An almost period of a function $F$ in a normed space is a shift $t$ for which $\Vert {\tau }_{t}F-F\Vert$ is small.

If finite sets $A,S$ in a group satisfy $|A+S|\le K|A|$, then for $q\ge 2$ and $0<ϵ<1$ there is $T\subseteq S$ with $|T|\ge (2K{)}^{-O(q/{ϵ}^2)}|S|$ such that every $t\in T-T$ is an $L^q$ almost period of $1_A\ast f$:

Sampling the normalized Fourier expansion of $f:{\mathbb{F}}_p^n\to \mathbb{C}$ gives an average of $O(q/{ϵ}^2)$ phase-adjusted characters whose $L^q$ error is at most $ϵ\Vert \hat{f}{\Vert }_1$. The common kernel of those characters is therefore a low-codimension subspace of almost periods of $f$.