A note on the largest sum-free sets of integers

Given $A$ a set of $N$ positive integers, an old question in additive combinatorics asks that whether $A$ contains a sum-free subset of size at least $N/3+\omega(N)$ for some increasing unbounded function $\omega$. The question is generally attacked in the literature by considering another conjecture, which asserts that as $N\to\infty$, $\max_{x\in\mathbb{R}/\mathbb{Z}}\sum_{n\in A}({\bf 1}_{(1/3,2/3)}-1/3)(nx)\to\infty$. This conjecture, if true, would also imply that a similar phenomenon occurs for $(2k,4k)$-sum-free sets for every $k\geq1$. In this note, we prove the latter result directly. The new ingredient of our proof is a structural analysis on the host set $A$, which might be of independent interest.