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

Source abstract

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 with a, n 1 ,. . ., n d ∈ $\mathbb{N}$ , 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 sum-free subset of size at least log n (log (3) n ) 1/32772 − o (1) .