Appendix to ‘Roth’s theorem on progressions revisited,’ by

Roth’s Theorem on Progressions Revisited

This paper is a sequel to [B]. Our main result is an improvement of the density condition for a subset A ⊂ {1,. .. , N } to contain a nontrivial arithmetic progression of length 3. More specifically, we prove the following Theorem 1. (0.1) δ ≫ (log log N) 2 (log N) 2/3 (N assumed sufficiently large), then A contains nontrivial progressions of length 3.

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.