Ergodic theorems with arithmetical weights

TL;DR

This paper proves that certain arithmetical functions, such as the divisor function, can serve as effective weights in the pointwise ergodic theorem, establishing almost everywhere convergence for dynamical systems.

Contribution

It introduces the use of arithmetical functions as weights in the ergodic theorem and proves their effectiveness using Bourgain's circle method.

Findings

Divisor function $d(n)$ is a good weighting function for the ergodic theorem.

Almost everywhere convergence established for weighted averages with arithmetical weights.

Results extend to functions like $ heta(n)$ and $J_s(n)$.

Abstract

We prove that the divisor function $d(n)$ counting the number of divisors of the integer $n$, is a good weighting function for the pointwise ergodic theorem. For any measurable dynamical system $(X, {\mathcal A},\nu,\tau)$ and any $f\in L^p(\nu)$, $p>1$, the limit $$ \lim_{n\to \infty}{1\over \sum_{k=1}^{n} d(k)} \sum_{k=1}^{n} d(k)f(\tau^k x)$$ exists $\nu$-almost everywhere. We also obtain similar results for other arithmetical functions, like $\theta(n)$ function counting the number of squarefree divisors of $n$ and the generalized Euler totient function $J_s(n)$, $s>0$. We use Bourgain's method, namely the circle method based on the shift model.