On the polynomials homogeneous ergodic bilinear averages with Liouville and M\"obius weights

TL;DR

This paper proves that certain bilinear averages weighted by Liouville or Möbius functions tend to zero almost surely, extending Bourgain's double recurrence theorem to include these multiplicative number theory weights.

Contribution

It generalizes Bourgain's double recurrence theorem by incorporating Liouville and Möbius weights into polynomial ergodic averages, a novel extension in ergodic theory.

Findings

Weighted averages with Liouville and Möbius functions converge to zero almost everywhere.

The result applies to non-constant polynomial iterates in ergodic systems.

Extends classical recurrence theorems to multiplicative number theory weights.

Abstract

We establish a generalization of Bourgain double recurrence theorem by proving that for any map $T$ acting on a probability space $(X,\mathcal{A},\mu)$, and for any non-constant polynomials $P, Q$ mapping natural numbers to themselves, for any $f,g \in L^2(X)$, and for almost all $x \in X$, we have $$\lim_{\bar{N} \longrightarrow +\infty} \frac{1}{N} \sum_{n=1}^{N}\boldsymbol{\nu}(n) f(T^{P(n)}x)g(T^{Q(n)}x)=0$$ where $\boldsymbol{\nu}$ is the Liouville function or the M\"{o}bius{\P} function.