Almost Arithmetic Progressions in the Primes and Other Large Sets

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 the Erdős conjecture on arithmetic progressions. 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.

• MSC: Primary 11B25
• Secondary 11N13
• 11B05

Acknowledgments

The author was financially supported by a Leverhulme Trust Research Fellowship (RF-2016-500) and an EPSRC Standard Grant (EP/R015104/1). He thanks Han Yu for many inspiring conversations related to the topics presented here and two anonymous referees for helpful comments.