Almost arithmetic progressions in the primes and other large sets

TL;DR

This paper presents a simple, self-contained argument showing that primes and large sets in the Erdős-Turán sense can approximate arbitrarily long arithmetic progressions, providing insight into their underlying arithmetic structure.

Contribution

It offers a straightforward heuristic approach demonstrating that primes and large sets get arbitrarily close to long arithmetic progressions, complementing the Green-Tao theorem.

Findings

Primes are arbitrarily close to long arithmetic progressions.

Large sets in Erdős-Turán sense exhibit similar arithmetic structure.

The proof is short, heuristic, and self-contained.

Abstract

A celebrated and deep result of Green and Tao states that the primes contain arbitrarily long arithmetic progressions. In this note I provide a straightforward argument demonstrating that the primes get arbitrarily close to arbitrarily long arithmetic progressions. The argument also applies to `large sets' in the sense of Erd\H{o}s-Tur\'an. The proof is short, completely self-contained, and aims to give a heuristic explanation of why the primes, and other large sets, possess arithmetic structure.