Roth's theorem for four variables and additive structures in sums of sparse sets

Abstract: We show that if a subset A of {1,...,N} does not contain any solutions to the equation x+y+z=3w with the variables not all equal, then A has size at most exp(-c(log N)^{1/7}) N, where c > 0 is some absolute constant. In view of Behrend's construction, this bound is of the right shape: the exponent 1/7 cannot be replaced by any constant larger than 1/2. We also establish a related result, which says that sumsets A+A+A contain long arithmetic progressions if A is a subset of {1,...,N}, or high-dimensional subspaces if A is a subset of a vector space over a finite field, even if A has density of the shape above.