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arXiv:2502.08624math.NT

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

Benjamin Bedert

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Abstract

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 $A'$ of size $|A'|\geqslant n/3+c\log \log n$. This answers a longstanding problem in additive combinatorics, originally due to Erdős.

arXiv comments
37 pages
Journal reference
DOI
Subjects
math.NTNumber Theory math.CAClassical Analysis and ODEs math.COCombinatorics

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  1. Graduate Student Solves Classic Problem About the Limits of Addition 1 comment

    • So cool. On the one hand, going from n/3 to (n/3 + log log n) seems like such a small improvement, but as the article shows, the history is formidable: - n/3 (Erdős, 1965) - (n+1)/3 (Alon and Kleitman, 1990) - (n+2)/3 (Bourgain, 1997) - n/3 + Ω(log log n) (this paper, Benjamin…

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