Furstenberg systems of bounded multiplicative functions and applications

TL;DR

This paper establishes a structural understanding of measure-preserving systems associated with multiplicative functions, showing they lack irrational spectrum and are composed of Bernoulli and nilsystems, with applications to disjointness and correlation properties.

Contribution

It provides a novel structural classification of systems linked to multiplicative functions, extending previous results and applying recent advances in Chowla's conjecture.

Findings

Multiplicative systems have no irrational spectrum.

Strongly aperiodic multiplicative functions satisfy a logarithmic disjointness conjecture.

Products of shifts of multiplicative functions do not correlate with zero-mean totally ergodic sequences.

Abstract

We prove a structural result for measure preserving systems naturally associated with any finite collection of multiplicative functions that take values on the complex unit disc. We show that these systems have no irrational spectrum and their building blocks are Bernoulli systems and infinite-step nilsystems. One consequence of our structural result is that strongly aperiodic multiplicative functions satisfy the logarithmically averaged variant of the disjointness conjecture of Sarnak for a wide class of zero entropy topological dynamical systems, which includes all uniquely ergodic ones. We deduce that aperiodic multiplicative functions with values plus or minus one have super-linear block growth. Another consequence of our structural result is that products of shifts of arbitrary multiplicative functions with values on the unit disc do not correlate with any totally ergodic deterministic sequence of zero mean. Our methodology is based primarily on techniques developed in a previous article of the authors where analogous results were proved for the M\"obius and the Liouville function. A new ingredient needed is a result obtained recently by Tao and Ter\"av\"ainen related to the odd order cases of the Chowla conjecture.