Published online by Cambridge University Press: 18 January 2018
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
(a) $R$ is divisible, i.e. $\overline{d}(R\cap u\mathbb{N})>0$ 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
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