On the homogeneous ergodic bilinear averages with $1$-bounded multiplicative weights

We establish a generalization of Bourgain double recurrence theorem and ergodic Bourgain-Sarnak's theorem by proving that for any aperiodic $1$-bounded multiplicative function $\boldsymbol{\nu}$, for any map $T$ acting on a probability space $(X,\mathcal{A},\mu)$, for any integers $a,b$, for any $f,g \in L^2(X)$, and for almost all $x \in X$, we have $$\frac{1}{N} \sum_{n=1}^{N} \boldsymbol{\nu}(n) f(T^{a n}x)g(T^{bn}x) \xrightarrow[N\rightarrow +\infty]{} 0.$$ We further present with proof the key ingredients of Bourgain's proof of his double recurrence theorem.