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 $(X,\mathcal{A},\nu )$, and $a,b\in \mathbb{Z}$, then for any $f,g\in L^2\left(X\right)$, for almost all $x\in X$,

$$\frac{1}{N}{\sum }_{n=1}^N\mathit{\nu }\left(n\right)f\left(T^{an}x\right)g\left(T^{bn}x\right)\underset{N\to +\infty }{\to }0,$$ where $\mathit{\nu }$ is the Liouville function or the Möbius function. We further obtain that the convergence almost everywhere holds for the short interval with the help of Zhan’s estimation. Moreover, we establish that if $T$ is weakly mixing and its restriction to its Pinsker algebra has singular spectrum, then for any integer $k\ge 1$, for any $f_j\in L^{\infty }\left(X\right)$, $j=1,\dots ,k$, for almost all $x\in X$, we have

$$\frac{1}{N}{\sum }_{n=1}^N\mathit{\nu }\left(n\right){\prod }_{j=1}^k{f}_j\left(T^{nj}x\right)\underset{N\to +\infty }{\to }0.$$