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

2012/10/05 by Dousse, Jehanne
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1210.1729

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 ≤ i ≤ j ≤ d} where a, n₁,..., 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, Szemerédi 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)}.

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