%0 Journal Article
%T Finding linear patterns of complexity one
%A Xuancheng Shao
%J Mathematics
%D 2013
%I arXiv
%X 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.
%U http://arxiv.org/abs/1309.0644v1