On the homogeneous ergodic bilinear averages with M\"{o}bius and liouville weights

It is shown that the homogeneous ergodic bilinear averages with M\"{o}bius 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 $x \in X$, $${\frac{1}{N}}\sum_{n=1}^{N} \nu(n) f(T^{an}x) g(T^{bn}x) \longrightarrow 0, \text{ as } N \longrightarrow +\infty, $$ 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 $${\frac{1}{N}} \sum_{n=1}^{N} \nu(n) \prod_{j=1}^{k}f_j({T_j}^nx) \longrightarrow 0, \text{ as } N \longrightarrow +\infty,$$ where $T_j$ are some powers of $T$, $j=1,\cdots,k$.