On the homogeneous ergodic bilinear averages with möbius and liouville weights


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.$

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Last time updated on May 20, 2019

This paper was published in HAL - Normandie Université.

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