Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics

Abstract

A set$R\subset \mathbb{N}$is calledrationalif 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 involverationally 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.