Roth’s Theorem: graph theoretical, analytic and combinatorial proofs

Abstract

In 1936, Erdős–Turán conjectured that any set of integers with positive upper density contains arbitrarily long arithmetic progressions. In 1953, Klaus Roth resolved this conjecture for progressions of length three. This theorem, known as Roth's Theorem, is the main topic of this thesis. In this dissertation we will understand, rewrite and collect some of the proofs of Roth's Theorem that have appeared over the years, while developing some of the problems that arise in each area. This includes the original Fourier analytic proof due to Roth (in a more modern language), the combinatorial proof due to Szemerédi, and finally, the graph theoretical proof based on Szemerédi's Regularity Lemma. We will also explore recent progress around this theorem, as the finite field analogue and the recent breakthrough concerning upper bounds for the cap set problem.