A set B is said to be sum-free if there are no x,y,z in B with x+y=z. A classical probabilistic argument of Erdös shows that any set of N integers contains a sum-free subset of size N/3, and this was later improved to (N+1)/3 by Alon and Kleitman, and then to (N+2)/3 by Bourgain using an elaborate Fourier-analytic approach. We show that there exists a constant c>0 such that any set of N integers contains a sum-free subset of size N/3+c log log N, confirming the longstanding suspicion that the 2/3 in Bourgain's bound can be improved to any large constant C (for large N). A key step in the proof consists of establishing inverse results giving combinatorial descriptions for sets of integers whose Fourier transform has small L^1-norm.
