# On the homogeneous ergodic bilinear averages with möbius and liouville weights

## Abstract

It is shown that the homogeneous ergodic bilinear averages with Möbius or Liouville weight converge almost surely to zero, that is, if T is a map acting on a probability space $(X, \mathcal{A}, \mu)$, and $a, b \in \mathbb{Z}$, then for any $f, g \in L^2(X)$, for almost all $x \in X$,$$\lim_{N \rightarrow +\infty}\frac1{N}\sum_{n=1}^{N}\nu(n) f(T^{an}x)g(T^{bn}x)=0,$$where $\nu$ is the Liouville function or the M\"{o}bius function. We further obtain thatthe convergence almost everywhere holds for the short interval with the help of Zhan's estimation. Also our proof yields a simple proof of Bourgain's double recurrence theorem.Moreover,we establish that if $T$ is weakly mixing and its restriction to its Pinsker algebra has singular spectrum, then for any integer $k \geq 1$, for any $f_j \in L^{\infty}(X),$ $j=1,\cdots,k$, for almost all $x \in X$, we have \lim_{N \rightarrow +\infty} \frac1{N}\sum_{n=1}^{N}\nu(n) \prod_{j=1}^{k}f(T^{nj}x)=0.\$

## Full text

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hal-01528209v2oai:HAL:hal-01528209v2
Last time updated on May 20, 2019

This paper was published in HAL - Normandie Université.

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