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 include many classical sets of number-theoretical origin such as the set of squarefree numbers, the set of abundant numbers, or sets of the form $\unicode[STIX]{x1D6F7}{x}:={n\in \mathbb{N}:\boldsymbol{\unicode[STIX]{x1D711}}(n)/n<x}$, where $x\in [0,1]$ and $\boldsymbol{\unicode[STIX]{x1D711}}$ is Euler’s totient function. We investigate the combinatorial and dynamical properties of rational sets and obtain new results in ergodic Ramsey theory. Among other things, we show that if $R\subset \mathbb{N}$ is a rational set with $\overline{d}(R)>0$, then the following are equivalent:(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 $$\begin{eqnarray}{n\in R:\overline{d}(E\cap (E-p_{1}(n))\cap \cdots \cap (E-p_{\ell }(n)))>\unicode[STIX]{x1D6FD}}\end{eqnarray}$$ 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.