A descriptive set theorist's proof of the pointwise ergodic theorem

TL;DR

This paper presents a combinatorial proof of the pointwise ergodic theorem for probability measure preserving actions of the integers, reducing the problem to a tiling problem that is straightforward for z.

Contribution

It introduces a novel combinatorial approach that simplifies the proof by reducing it to a tiling problem, offering potential extensions to other groups.

Findings

Provides a short combinatorial proof of the ergodic theorem

Reduces the theorem to a tiling problem for z

Suggests possible extensions to other groups and tiles

Abstract

We give a short combinatorial proof of the classical pointwise ergodic theorem for probability measure preserving $\mathbb{Z}$-actions. Our approach reduces the theorem to a tiling problem: tightly tile each orbit by intervals with desired averages. This tiling problem is easy to solve for $\mathbb{Z}$ with intervals as tiles. However, it would be interesting to find other classes of groups and sequences of tiles for which this can be done, since then our approach would yield a pointwise ergodic theorem for such classes.