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

Let $G={\mathbb{F}}_p^n$, $m\ge 1$ be an integer, and $\epsilon >0$. Use counting convolution,and the counting Lp norm $\Vert F{\Vert }_{{\ell }^{2m}}=({\sum }_x|F(x){|}^{2m}{)}^{1/(2m)}$. There is a vector subspace $V\le G$ of codimension of a vector subspace $O(m{\epsilon }^{-2})$ such that every $t\in V$ satisfiesEquivalently, with normalized convolution on a finite group $f=F/|G|$, normalized Lp norms and subset density $\alpha =|A|/|G|$, the error is $\epsilon \alpha$.

For nonempty $A$, Parseval identity gives ${\sum }_{\chi }|\widehat{1_A}(\chi ){|}^2=\alpha$, so the Fourier coefficients on a finite abelian group of $f$ have total absolute sum $\alpha$. Sampling $O(m{\epsilon }^{-2})$ phase-adjusted characters, using the Marcinkiewicz–Zygmund inequality, approximates $f$ in normalized $L^{2m}$ within $\epsilon \alpha /2$. The common kernel of these characters has at most that codimension of a vector subspace. The approximation is unchanged by every shift in that kernel; the triangle inequality gives the stated error for $f$. If $A$ is empty, take $V=G$.

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]

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 $f$.