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arXiv · 2008.04886

On the polynomials homogeneous ergodic bilinear averages with Liouville and Möbius weights

el Houcein el Abdalaoui
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Abstract

We establish a generalization of Bourgain double recurrence theorem by proving that for any map $T$ acting on a probability space $(X,\mathcal{A},μ)$, and for any non-constant polynomials $P, Q$ mapping natural numbers to themselves, for any $f,g \in L^2(X)$, and for almost all $x \in X$, we have $$\lim_{\bar{N} \longrightarrow +\infty} \frac{1}{N} \sum_{n=1}^{N}\boldsymbolν(n) f(T^{P(n)}x)g(T^{Q(n)}x)=0$$ where $\boldsymbolν$ is the Liouville function or the Möbius¶ function.

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el Houcein el Abdalaoui. 2020-08-10. On the polynomials homogeneous ergodic bilinear averages with Liouville and Möbius weights. https://arxiv.org/abs/2008.04886

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