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

Abstract

We establish a generalization of Bourgain double recurrence theorem by proving that for any map $T$ acting on a probability space $(X,\mathcal{A},\mu )$, 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 $$\underset{\bar{N}⟶+\infty }{\lim }\frac{1}{N}\sum _{n=1}^N\mathbf{\nu }(n)f(T^{P(n)}x)g(T^{Q(n)}x)=0$$ where $\mathbf{\nu }$ is the Liouville function or the Möbius¶ function.

18 pages. We generalize the first main result (Theorem 4.1) in arXiv:1706.07280 to the polynomial cases and we will fill the gap

References in corpus (3)

• Pointwise double recurrence and nilsequences
• A uniform nilsequence Wiener-Wintner theorem for bilinear ergodic averages
• Simple proof of Bourgain bilinear ergodic theorem and its extension to polynomials and polynomials in primes