A quantitative ergodic theory proof of Szemer\'edi's theorem

TL;DR

This paper presents a self-contained, quantitative ergodic theory proof of Szemerédi's theorem that avoids advanced set-theoretic and harmonic analysis tools, providing explicit bounds and making the proof more elementary.

Contribution

It offers a new, elementary, and quantitative ergodic theory proof of Szemerédi's theorem, removing the need for the axiom of choice and inverse theorems.

Findings

Provides explicit quantitative bounds for Szemerédi's theorem.

Develops an elementary ergodic theory proof avoiding advanced set theory.

The proof is self-contained and does not rely on Fourier analysis or inverse theorems.

Abstract

A famous theorem of Szemer\'edi asserts that given any density $0 < \delta \leq 1$ and any integer $k \geq 3$, any set of integers with density $\delta$ will contain infinitely many proper arithmetic progressions of length $k$. For general $k$ there are essentially four known proofs of this fact; Szemer\'edi's original combinatorial proof using the Szemer\'edi regularity lemma and van der Waerden's theorem, Furstenberg's proof using ergodic theory, Gowers' proof using Fourier analysis and the inverse theory of additive combinatorics, and Gowers' more recent proof using a hypergraph regularity lemma. Of these four, the ergodic theory proof is arguably the shortest, but also the least elementary, requiring in particular the use of transfinite induction (and thus the axiom of choice), decomposing a general ergodic system as the weakly mixing extension of a transfinite tower of compact extensions. Here we present a quantitative, self-contained version of this ergodic theory proof, and which is ``elementary'' in the sense that it does not require the axiom of choice, the use of infinite sets or measures, or the use of the Fourier transform or inverse theorems from additive combinatorics. It also gives explicit (but extremely poor) quantitative bounds.