Large sum-free subsets of sets of integers via $L^1$-estimates for trigonometric series

A set $B$ is said to be \emph{sum-free} if there are no $x,y,z\in B$ with $x+y=z$. We show that there exists a constant $c>0$ such that any set $A$ of $n$ integers contains a sum-free subset $A^{\mathrm{\prime }}$ of size $\mathrm{\mid }{A}^{\mathrm{\prime }}\mathrm{\mid }⩾n\mathrm{/}3+c\log \log n$. This answers a longstanding problem in additive combinatorics, originally due to Erdős.