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 $x\in X$, $$\frac{1}{N}\sum _{n=1}^N\nu (n)f(T^{an}x)g(T^{bn}x)⟶0,\text{ as }N⟶+\mathrm{\infty },$$ 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 $k\ge 1$, for any $f_j\in L^{\mathrm{\infty }}(X)$, $j=1,\cdots ,k,$ for almost all $x\in X$, we have $$\frac{1}{N}\sum _{n=1}^N\nu (n)\prod _{j=1}^k{f}_j({T_j}^{n}x)⟶0,\text{ as }N⟶+\mathrm{\infty },$$ where $T_j$ are some powers of $T$, $j=1,\cdots ,k$.