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 $\mathit{\nu }$, for any map $T$ acting on a probability space $(X,\mathcal{A},\mu )$, for any integers $a,b$, for any $f,g \in L^2(X)$,andforalmostall$x \in X$, we have $$\frac{1}{N}\sum _{n=1}^N\mathit{\nu }(n)f(T^{an}x)g(T^{bn}x)\underset{N\to +\mathrm{\infty }}{\overset{}{\to }}0.$$ We further present with proof the key ingredients of Bourgain’s proof of his double recurrence theorem.