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April 2020 On the homogeneous ergodic bilinear averages with Möbius and Liouville weights
E. H. el Abdalaoui
Illinois J. Math. 64(1): 1-19 (April 2020). DOI: 10.1215/00192082-8165574

Abstract

It is shown that the homogeneous ergodic bilinear averages with Möbius or Liouville weight converge almost surely to zero; that is, if T is a map acting on a probability space (X,A,ν), and a,bZ, then for any f,gL2(X), for almost all xX,

1Nn=1Nν(n)f(Tanx)g(Tbnx)N+0, where ν is the Liouville function or the Möbius function. We further obtain that the convergence almost everywhere holds for the short interval with the help of Zhan’s estimation. Moreover, we establish that if T is weakly mixing and its restriction to its Pinsker algebra has singular spectrum, then for any integer k1, for any fjL(X), j=1,,k, for almost all xX, we have

1 N n = 1 N ν ( n ) j = 1 k f j ( T n j x ) N + 0 .

Citation

Download Citation

E. H. el Abdalaoui. "On the homogeneous ergodic bilinear averages with Möbius and Liouville weights." Illinois J. Math. 64 (1) 1 - 19, April 2020. https://doi.org/10.1215/00192082-8165574

Information

Received: 14 November 2018; Revised: 25 October 2019; Published: April 2020
First available in Project Euclid: 6 March 2020

zbMATH: 07179187
MathSciNet: MR4072639
Digital Object Identifier: 10.1215/00192082-8165574

Subjects:
Primary: 37A30
Secondary: 11B30, 11N37, 28D05, 37A45, 5D10

Rights: Copyright © 2020 University of Illinois at Urbana-Champaign

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Vol.64 • No. 1 • April 2020
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