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 that the 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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https://hal.archives-ouvertes.fr/hal-01528209
Contributeur : El Houcein El Abdalaoui <>
Soumis le : mardi 13 juin 2017 - 21:58:43
Dernière modification le : vendredi 16 juin 2017 - 01:05:41

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  • HAL Id : hal-01528209, version 2

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El Houcein El Abdalaoui. On the homogeneous ergodic bilinear averages with möbius and liouville weights. 2017. <hal-01528209v2>

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