Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics

TL;DR

This paper explores the properties of rational sets in number theory and their implications in ergodic Ramsey theory and symbolic dynamics, establishing new results on polynomial recurrence and rationally almost periodic sequences.

Contribution

It introduces new equivalences for rational sets with positive density related to polynomial recurrence and demonstrates their applications in symbolic dynamics and ergodic theory.

Findings

Rational sets with positive density are equivalent to being divisible and serving as averaging sets for polynomial recurrence.

Rational and divisible sets guarantee positive density intersections in polynomial recurrence contexts.

Rationally almost periodic sequences are generic points for ergodic measures with rational discrete spectrum.

Abstract

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.