Arithmetic progressions and the primes - El Escorial lectures

TL;DR

This paper reviews recent advances in understanding how primes and dense sets contain arbitrarily long arithmetic progressions, highlighting the mathematical machinery used in these proofs.

Contribution

It explains the key tools and methods behind recent proofs of infinitely many arithmetic progressions in primes and dense integer sets.

Findings

Existence of arbitrarily long arithmetic progressions in primes

Methods applicable to dense subsets of integers

Overview of recent progress in additive number theory

Abstract

We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length $k$ in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes.