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.

Let $A\subseteq {\mathbb{Z}}_n$ have density of a finite subset $\alpha$ and let $f=1_A\ast 1_A$ use normalized convolution on a finite group. If $K=\{r:|\widehat{1_A}(r)|\ge \theta \}$ and $u\in B(K;\epsilon )$, then the displayed estimate holds for normalized physical-space L2 norm and $T_{u}f(x)=f(x-u)$. The Bohr set controls the translation factors on the large spectrum; the fourth Fourier moment bound for an indicator function controls their total weight. Outside $K$, use $|1-e^{2\pi iru/n}|\le 2$ and Parseval identity on a finite group.

For $f:G\to \mathbb{C}$ and $X\ge 0$, an $L^p$ almost period of $f$ with error $X$ is an element $t\in G$ such that

If $A\subseteq {\mathbb{F}}_p^n$, $m\ge 1$, and $\epsilon >0$, then the $L^{2m}$ almost periods of $1_A\ast 1_A$ with error $\epsilon |A|p^{n/(2m)}$ contain a vector subspace of codimension $O(m{\epsilon }^{-2})$. One proof samples the Fourier expansion of the convolution using the Marcinkiewicz–Zygmund inequality; the sampled characters have a common kernel of the required codimension.

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 $f$.