A note on the largest sum-free sets of integers

Abstract

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 $N\to \infty$, ${\max }_{x\in \mathbb{R}/\mathbb{Z}}{\sum }_{n\in A}(1_{(1/3,2/3)}-1/3)(nx)\to \infty$. This conjecture, if true, would also imply that a similar phenomenon occurs for $(2k,4k)$-sum-free sets for every $k\ge 1$. 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.

21 pages, to appear in J. Lond. Math. Soc