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.