On the homogeneous ergodic bilinear averages with $1$-bounded multiplicative weights
el Houcein el Abdalaoui
TL;DR
The paper extends Bourgain's double recurrence framework to homogeneous bilinear ergodic averages weighted by an aperiodic 1-bounded multiplicative function ν, proving that $\frac{1}{N}\sum_{n=1}^{N} ν(n) f(T^{a n}x) g(T^{b n}x) \to 0$ for a.e. $x$ when $f,g \in L^2(X)$ and $T$ is ergodic. The approach combines a Wiener–Wintner refinement of the Daboussi–Katai–Bourgain–Sarnak–Ziegler criterion with Gowers-Host-Kra seminorms and Bourgain’s harmonic analysis methods, using Calderón transference to move between discrete and dynamical settings. Key contributions include a detailed WW-DKBSZ framework, the integration of Gowers norms into the dynamical bilinear setting, and a proof strategy that mirrors Bourgain’s, including LP-type localizations and spectral decompositions. As a consequence, the Möbius and Liouville functions are shown to be good weights for these bilinear averages in the appropriate ergodic contexts, extending the scope of ergodic theorems with arithmetic weights and enriching the interplay between number theory and ergodic theory.
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 $\boldsymbolν$, for any map $T$ acting on a probability space $(X,\mathcal{A},μ)$, 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.