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.Comment: 23 page