Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics
Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics
A set $R\subset \mathbb{N}$ is called rational if it is well approximable by finite unions of arithmetic progressions, meaning that for every $\unicode[STIX]{x1D716}>0$ there exists a set $B=\bigcup _{i=1}^{r}a_{i}\mathbb{N}+b_{i}$ , where $a_{1},\ldots ,a_{r},b_{1},\ldots ,b_{r}\in \mathbb{N}$ , such that $$\begin{eqnarray}\overline{d}(R\triangle B):=\limsup _{N\rightarrow \infty }\frac{|(R\triangle B)\cap \{1,\ldots ,N\}|}{N}<\unicode[STIX]{x1D716}.\end{eqnarray}$$ Examples of rational sets …