A cubic nonconventional ergodic average with multiplicative or Mangoldt weights

TL;DR

This paper proves convergence results for cubic nonconventional ergodic averages with multiplicative or Mangoldt weights under certain conditions, extending ergodic theory with number-theoretic weights.

Contribution

It establishes almost sure convergence of cubic ergodic averages with multiplicative and Mangoldt weights, under Daboussi-Delange and nilsystem assumptions, with quantitative bounds.

Findings

Convergence of cubic averages with multiplicative weights under Daboussi-Delange condition.

Quantitative bounds involving logarithmic decay for self-correlation averages.

Almost sure convergence of ergodic averages with Mangoldt weights on nilsystems.

Abstract

We show that the cubic nonconventional ergodic averages of any order with a bounded multiplicative function weight converge almost surely to zero provided that the multiplicative function satisfies a strong Daboussi-Delange condition. We further obtain that the Ces\`{a}ro mean of the self-correlations and some moving average of the self-correlations of such multiplicative functions converge to zero. Our proof gives, for any $N \geq 2$, $$\frac1{N}\sum_{m=1}^{N}\Big|\frac1{N}\sum_{n=1}^{N} \bnu(n) \bnu(n+m)\Big| \leq \frac{C}{\log(N)^{\epsilon}},$$ and $$\frac1{N^2}\sum_{n,p=1}^{N}\Big|\frac1{N}\sum_{m=1}^{N} \bnu(m) \bnu(n+m)\bnu(m+p)\bnu(n+m+p)\Big| \leq \frac{C}{\log(N)^{\varepsilon}},$$ where $C,\varepsilon$ are some positive constants and $\bnu$ is a bounded multiplicative function satisfying a Daboussi-Delange condition with logarithmic speed. We further establish that the cubic nonconventional ergodic averages of any order with Mangoldt weight converge almost surely provided that all the systems are nilsystems.