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

摘要

We prove a generalisation of Roth's theorem for arithmetic progressions to d-configurations, which are sets of the form {ni +nj +a}1 ≤ i ≤ j ≤ d with a, n 1,...,nd 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 logn(log ~((3)) n)~(1/32772-o(1)).