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

2020/12/09 by El Houcein El Abdalaoui, Abdalaoui, el Houcein el
Mathematics · #Limits and Structures in Graph Theory #Analytic Number Theory Research #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2012.06323

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,A,μ), for any integers a,b, for any f,g ∈ L²(X), and for almost all x ∈ X, we have (1)/(N) ∑_n=1^N \boldsymbolν(n) f(T^{a n}x)g(T^bnx) \xrightarrow[N→ +∞] 0. We further present with proof the key ingredients of Bourgain's proof of his double recurrence theorem.

Citations

• Pointwise double recurrence and nilsequences
• On Multiplicative Arithmetical Functions whose Modulus does not Exceed One

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