Ergodic theorems with arithmetical weights

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 $$\underset{n\to \infty }{\lim }\frac{1}{{\sum }_{k=1}^{n}d(k)}\sum _{k=1}^{n}d(k)f({\tau }^{k}x)$$ 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.

25 pages