Finding linear patterns of complexity one

TL;DR

This paper extends Roth's theorem to s-configurations, showing that sets with sufficient density contain these configurations, and improves results related to sum-free subsets.

Contribution

It generalizes Roth's theorem to s-configurations and establishes density thresholds for their existence, also improving bounds on sum-free subsets.

Findings

Sets with density at least (log N)^{-c(s)} contain nontrivial s-configurations

Provides a density condition guaranteeing the presence of s-configurations

Improves bounds on sum-free subset problems

Abstract

We study the following generalization of Roth's theorem for 3-term arithmetic progressions. For s>1, define a nontrivial s-configuration to be a set of s(s+1)/2 integers consisting of s distinct integers x_1,...,x_s as well as all the averages (x_i+x_j)/2. Our main result states that if a set A contained in {1,2,...,N} has density at least (log N)^{-c(s)} for some positive constant c(s)>0 depending on s, then A contains a nontrivial s-configuration. We also deduce, as a corollary, an improvement of a problem involving sumfree subsets.