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On a generalisation of Roth's theorem for arithmetic progressions and applications to sum-free subsets (1210.1729v2)
Published 5 Oct 2012 in math.NT and math.CO
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)}.