A set $R\subset \mathbb{ N}$ is called rational if it is well-approximable by finite unions of arithmetic progressions. 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 $\Phi_x:=\{ n\in\mathbb{ N}: \frac{ \boldsymbol{ \varphi}(n)}{ n}<x\}$, where $x\in[0,1]$ and $\boldsymbol{ \varphi}$ is Euler's totient function. We investigate the combinatorial and dynamical properties of rational sets and obtain new results in ergodic Ramsey theory. We show that if $R$ 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 $R$ 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 $\beta>0$ such that the set $$\{ n\in R:\overline{ d}( E\cap (E-p_1(n))\cap\ldots\cap(E-p_\ell(n)))>\beta\}$$ has positive lower density. Ramsey-theoretical applications naturally lead to problems in symbolic dynamics, which involve rationally almost periodic sequences. We prove that if $\mathcal{ A}$ is a finite alphabet, $\eta\in\mathcal{ A}^\mathbb{ N}$ is rationally almost periodic, $S$ denotes the left-shift on $\mathcal{ A}^\mathbb{ Z}$ and $$X:=\{ y\in \mathcal{ A}^\mathbb{ Z} : \text{ each finite word appearing in $y$ appears in }\eta\},$$ then $\eta$ is a generic point for an $S$-invariant probability measure $\nu$ on $X$ such that $(X,\nu,S)$ is ergodic and has rational discrete spectrum.
Categories: math.DS