= Syndetic near returns of an almost periodic observable
{title2=$R_\varepsilon=\{n\geq1:\|U^nf-f\|_2<\varepsilon\}$}
If the forward orbit of $f$ under an <isometry> $U$ is totally bounded, choose a finite net whose centers are $U^{n_j}f$ and let $M=\max_jn_j$. For every sufficiently large $n$, some center is within $\varepsilon$ of $U^nf$, so cancellation gives $n-n_j\in R_\varepsilon$. Thus $R_\varepsilon$ is a <syndetic set>. Telescoping also gives $\|U^{kn}f-f\|_2\leq k\|U^nf-f\|_2$.
Back to article page
