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

Abstract

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 $\hat{\mathbf{I}}½$, for any map $T$ acting on a probability space $(X,\mathcal{A},\hat{I}¼)$, for any integers $a,b$, for any $f,g\in L^2(X)$, and for almost all $x\in X$, we have \[\frac{1}{N} \sum_{n=1}^{N} \boldsymbolν(n) f(T^{a n}x)g(T^{bn}x) \xrightarrow[N\rightarrow +\infty]{} 0.\] We further present with proof the key ingredients of Bourgain's proof of his double recurrence theorem.

25 pages, 35 references and 7 lemmas. Scientific Comments are welcome. In this revised version, the proof of the main theorem (Theorem 5.1) is revised and augmented. arXiv admin note: text overlap with arXiv:2008.04886