On the homogeneous ergodic bilinear averages with $1$-bounded multiplicative weights

TL;DR

This paper proves a generalized ergodic theorem involving multiplicative weights, extending Bourgain's double recurrence theorem and the ergodic Bourgain-Sarnak's theorem, with implications for aperiodic multiplicative functions.

Contribution

It establishes a new convergence result for bilinear averages with multiplicative weights, generalizing previous theorems in ergodic theory and number theory.

Findings

Convergence of bilinear averages with multiplicative weights to zero

Extension of Bourgain's double recurrence theorem

Proof of key ingredients of Bourgain's original proof

Abstract

We establish a generalization of Bourgain double recurrence theorem and ergodic Bourgain-Sarnak's theorem by proving that for any aperiodic $1$-bounded multiplicative function $\boldsymbol{\nu}$, for any map $T$ acting on a probability space $(X,\mathcal{A},\mu)$, for any integers $a,b$, for any $f,g \in L^2(X)$, and for almost all $x \in X$, we have \[\frac{1}{N} \sum_{n=1}^{N} \boldsymbol{\nu}(n) f(T^{a n}x)g(T^{bn}x) \xrightarrow[N\rightarrow +\infty]{} 0.\] We further present with proof the key ingredients of Bourgain's proof of his double recurrence theorem.