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}0$ for all $u\in \mathbb{N}$;(b) $R$ is an averaging set of polynomial single recurrence;(c) $R$ is an averaging set of polynomial multiple recurrence.As an application, we show that if $R\subset \mathbb{N}$ is rational and divisible, then for any set $E\subset \mathbb{N}$ with $\overline{d}(E)>0$ and any polynomials $p_{i}\in \mathbb{Q}[t]$, $i=1,\ldots ,\ell$, which satisfy $p_{i}(\mathbb{Z})\subset \mathbb{Z}$ and $p_{i}(0)=0$ for all $i\in \{1,\ldots ,\ell \}$, there exists $\unicode[STIX]{x1D6FD}>0$ such that the set \begin{eqnarray}{n\in R:\overline{d}(E\cap (E-p_{1}(n))\cap \cdots \cap (E-p_{\ell }(n)))>\unicode[STIX]{x1D6FD}}\end{eqnarray}has positive lower density.Ramsey-theoretical applications naturally lead to problems in symbolic dynamics, which involve rationally almost periodic sequences (sequences whose level-sets are rational). We prove that if ${\mathcal{A}}$ is a finite alphabet, $\unicode[STIX]{x1D702}\in {\mathcal{A}}^{\mathbb{N}}$ is rationally almost periodic, $S$ denotes the left-shift on ${\mathcal{A}}^{\mathbb{Z}}$ and\begin{eqnarray}X:={y\in {\mathcal{A}}^{\mathbb{Z}}:\text{each word appearing in}y\text{appears in}~\unicode[STIX]{x1D702}},\end{eqnarray}$$ then $\unicode[STIX]{x1D702}$ is a generic point for an $S$-invariant probability measure $\unicode[STIX]{x1D708}$ on $X$ such that the measure-preserving system $(X,\unicode[STIX]{x1D708},S)$ is ergodic and has rational discrete spectrum.