A structure theorem for level sets of multiplicative functions and applications

TL;DR

This paper establishes a structure theorem for level sets of multiplicative functions, linking their properties to polynomial recurrence and refining results like the polynomial Szemerédi theorem.

Contribution

It introduces a decomposition of level sets into almost periodic and pseudo-random parts, enabling new results on polynomial recurrence and density properties.

Findings

Level sets of multiplicative functions with positive density are divisible.

Such sets are averaging sets for polynomial multiple recurrence.

Refinement of polynomial Szemerédi theorem for multiplicative function level sets.

Abstract

Given a level set $E$ of an arbitrary multiplicative function $f$, we establish, by building on the fundamental work of Frantzikinakis and Host [13,14], a structure theorem which gives a decomposition of $\mathbb{1}_E$ into an almost periodic and a pseudo-random parts. Using this structure theorem together with the technique developed by the authors in [3], we obtain the following result pertaining to polynomial multiple recurrence. Let $E=\{n_1<n_2<\ldots\}$ be a level set of an arbitrary multiplicative function with positive density. Then the following are equivalent: - $E$ is divisible, i.e. the upper density of the set $E\cap u\mathbb{N}$ is positive for all $u\in\mathbb{N}$; - $E$ is an averaging set of polynomial multiple recurrence, i.e. for all measure preserving systems $(X,\mathcal{B},\mu,T)$, all $A\in\mathcal{B}$ with $\mu(A)>0$, all $\ell\geq 1$ and all polynomials $p_i\in\mathbb{Z}[x]$, $i=1,\ldots,\ell$, with $p_i(0)=0$ we have $$ \lim_{N\to\infty}\frac{1}{N}\sum_{j=1}^N \mu\big(A\cap T^{-p_1(n_j)}A\cap\ldots\cap T^{-p_\ell(n_j)}A\big)>0. $$ We also show that if a level set $E$ of a multiplicative function has positive upper density, then any self-shift $E-r$, $r\in E$, is a set of averaging polynomial multiple recurrence. This in turn leads to the following refinement of the polynomial Szemer\'edi theorem (cf. [4]). Let $E$ be a level set of an arbitrary multiplicative function, suppose $E$ has positive upper density and let $r\in E$. Then for any set $D\subset \mathbb{N}$ with positive upper density 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 $$ \left\{\,n\in E-r:\overline{d}\Big(D\cap (D-p_1(n))\cap \ldots\cap(D-p_\ell(n)) \Big)>\beta \,\right\} $$ has positive lower density.