Roth’s Theorem on Progressions Revisited

Roth’s Theorem on 3-term Arithmetic Progressions

This article is a discussion about the proof of a classical theorem of Roth’s regarding the existence of three term arithmetic progressions in certain subsets of the integers. Before beginning with this task, however, we will take a brief look at the history and motivation behind Roth’s theorem. The questions and ideas surrounding this subject may have begun with a wonderful theorem due to van ...

On a generalisation of Roth’s theorem for arithmetic progressions and applications to sum-free subsets

We prove a generalisation of Roth’s theorem for arithmetic progressions to d-configurations, which are sets of the form {ni+nj +a}1≤i≤j≤d with a, n1, ..., nd ∈ N, using Roth’s original density increment strategy and Gowers uniformity norms. Then we use this generalisation to improve a result of Sudakov, Szemerédi and Vu about sum-free subsets [10] and prove that any set of n integers contains a...

Roth’s Theorem on Arithmetic Progressions

A quantitative improvement for Roth's theorem on arithmetic progressions

We improve the quantitative estimate for Roth’s theorem on threeterm arithmetic progressions, showing that if A ⊂ {1, . . . , N} contains no nontrivial three-term arithmetic progressions then |A| N(log logN)4/ logN . By the same method we also improve the bounds in the analogous problem over Fq [t] and for the problem of finding long arithmetic progressions in a sumset.

An Introduction to Additive Combinatorics

This is a slightly expanded write-up of my three lectures at the Additive Combinatorics school. In the first lecture we introduce some of the basic material in Additive Combinatorics, and in the next two lectures we prove two of the key background results, the Freiman-Ruzsa theorem and Roth’s theorem for 3-term arithmetic progressions. Lecture I: Introductory material 1. Basic Definitions. 2. I...