A set B is said to be sum-free if there are no x, y, z ∈ B with x + y = z. A classical probabilistic argument of Erdős shows that any set of N integers contains a sum-free subset of size N/3, and this was later improved 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, answering a longstanding problem in additive combinatorics. A key step in the proof consists of establishing inverse results giving combinatorial descriptions for sets of integers whose Fourier transform has small L¹-norm.