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

TL;DR

This paper extends Roth's theorem to d-configurations using advanced combinatorial methods and applies it to show that large sets of integers necessarily contain relatively large sum-free subsets.

Contribution

It generalizes Roth's theorem for arithmetic progressions to d-configurations and improves bounds on the size of sum-free subsets in large integer sets.

Findings

Generalization of Roth's theorem to d-configurations

Improved lower bounds for sum-free subsets in large sets

Application of Gowers norms and density increment strategy

Abstract

We prove a generalisation of Roth's theorem for arithmetic progressions to d-configurations, which are sets of the form {n_i+n_j+a}_{1 \leq i \leq j \leq d} where a, n_1,..., n_d are nonnegative integers, using Roth's original density increment strategy and Gowers uniformity norms. Then we use this generalisation to improve a result of Sudakov, Szemer\'edi and Vu about sum-free subsets and prove that any set of n integers contains a sum-free subset of size at least log n (log log log n)^{1/32772 - o(1)}.