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

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)}.

Categories: math.NT math.CO