Bedert 2025 large sum free subsets sets integers

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theorem_1_2: Bedert's lower bound for the largest sum-free subset of a set of n integers, n/3 + c log log n, the first improvement of Erdős's n/3 by an unbounded term and the answer to Problem 1 of Green's list; an unrefereed preprint.

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Benjamin Bedert, Large sum-free subsets of sets of integers via $L^1$-estimates for trigonometric series. arXiv:2502.08624, doi:10.48550/arXiv.2502.08624 (2025).

The retained folder-name PDF is arXiv:2502.08624v1 (12 February 2025; 37 pages), the only arXiv version on 2026-09-18, with no journal reference on arXiv and no Crossref record: an unrefereed preprint. Before 2026-09-18 this digest was written from the arXiv abstract alone; it was rewritten from the PDF on that date. Read status: claims checked for the definitions of sum-free, $S(A)$ and $S(N)$ (display (1), p. 1), Problem 1.1, Theorem 1.2 and Theorem 1.3 (p. 2) and Theorem 2.2 (p. 3), each read clause by clause in the text layer, with the overview of Section 2 (pp. 3--5); the proof (Sections 4--9, pp. 7--34) was not read. The main statement is on theorem_1_2. The arXiv record (https://arxiv.org/abs/2502.08624, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

A set $B$ is sum-free when no $x,y,z\in B$ satisfy $x+y=z$ (equal $x$ and $y$ not excluded); $S(A)$ is the largest size of a sum-free subset of $A$, and $S(N)$ is the minimum of $S(A)$ over sets of $N$ positive integers (p. 1). The introduction recalls Erdős's rotation argument giving $S(A)\ge|A|/3$ for $A\subset\mathbb Z\setminus\{0\}$, the improvements $S(N)\ge(N+1)/3$ (Alon and Kleitman) and $S(N)\ge(N+2)/3$ (Bourgain, "using an elaborate Fourier analytic approach"; Shakan's alternative proof), and states Problem 1.1, "Is there a function $\omega(N)\to\infty$ such that $S(N)\ge\frac N3+\omega(N)$?", "listed as Problem 1 on Green's list [8] of 100 open problems". Theorem 1.2 answers it: there is $c>0$ such that $S(A)\ge|A|/3+c\log\log|A|$ for all finite $A\subset\mathbb Z$, in particular $S(N)\ge N/3+c\log\log N$. Theorem 1.3, a "99% Structure Theorem", says that a set $A\subset\mathbb Z\setminus\{0\}$ of size $N$ with $S(A)\le N/3+C$ has a Freiman-isomorphic copy inside $[-N^{C^{O(1)}},N^{C^{O(1)}}]$ and, for any $K>1$, a partition into sets $A_j$ of size $\gg(KC)^{-O(1)}N$ with doubling $|A_j-A_j|\le(CK)^{O(1)}|A_j|$, each inside a generalized arithmetic progression of dimension $\ll(KC)^{O(1)}$, plus a remainder of size $\ll(KC)^{-10}N$. A table on p. 2 lists the upper-bound constants $c$ with $S(N)\le cN+o(N)$: $7/15$ (Hinton), $3/7$ (Klarner), $12/29$ (Alon and Kleitman), $2/5$ (Malouf; Füredi), $11/28$ (Lewko), $11/28-\varepsilon$ (Alon) and $1/3$ (Eberhard, Green and Manners), the last by the arithmetic regularity lemma with an ineffective $o(N)$, Eberhard later giving explicit examples.

Section 2 sets up the proof: since $0$ lies in no sum-free set it may be removed, and with $\varphi$ the indicator of $(1/3,2/3)$ on $\mathbb R/\mathbb Z$, $S(A)\ge N/3+\max_x\sum_{a\in A}(\varphi-\tfrac13)(ax)$, the quantity Bourgain bounded below by $1/3$ through the Fourier series $F_A(x)=\sum_{a\in A}\sum_{n\ge1}\frac{\chi(n)}n\cos2\pi nax$ ($\chi$ a character modulo $3$). Theorem 2.2 (p. 3), which implies Theorem 1.2: for $A\subset\mathbb Z\setminus\{0\}$ there is an $F_4$-isomorphic $B\subset\mathbb Z\setminus\{0\}$ (Definition 2.1: a bijection preserving all relations $\sum_{i\le4}\varepsilon_ib_i=0$, $\varepsilon_i\in\{-1,0,1\}$) with $\max_x\sum_{b\in B}(\varphi-\tfrac13)(bx)\gg\log\log|B|$. The route (pp. 4--5): Bourgain's observation that $\|F_A\|_1\gg C$ follows from $\|\hat1_A\|_1\gg C\log N$; inverse theorems for sets with small $L^1$ norm (Section 5, small additive dimension); a dense Freiman-isomorphic model (Section 6); the distribution of $A$ modulo powers of primes $p\le(\log N)^{1/2}$ (Section 7, Proposition 7.9: either $\|F_A\|_1\gg\log\log N$ or the distribution is highly structured); non-Archimedean test functions (Section 8); and the global structure of sets with $S(A)\le N/3+C$ (Section 9). Bourgain's asymmetric-interval method for $(3,1)$-sum-free sets ($S_{(3,1)}(N)\ge N/4+(\log N)^{1-o(1)}$) and its extension by Jing and Wu are recalled as inapplicable to $S(N)$, since $(1/3,2/3)$ is the unique sum-free interval of measure $1/3$.

For problem 792 the theorem is the site's best lower bound, $f(n)\ge n/3+c\log\log n$. For problem 790 it does not apply: that problem forbids an element equal to a sum of two or more distinct other elements, a condition that the two-term sum-free property treated here does not imply, and the paper says nothing about it.

Source: https://arxiv.org/abs/2502.08624.

Bears on. #792: Theorem 1.2 (p. 2) is the site's lower bound $f(n)\ge n/3+c\log\log n$, the answer to Problem 1 of Green's list, recorded with the preprint qualification. #790: not applicable; the two-term sum-free condition treated here does not imply that problem's condition on sums of two or more distinct summands.

Results to transcribe.

• Theorem 1.2 (p. 2): There is c > 0 such that every finite set A of integers has a sum-free subset of size at least |A|/3 + c log log |A|; in particular S(N)

  = N/3 + c log log N.

• Theorem 1.3 (p. 2): A set A of N nonzero integers with S(A) <= N/3 + C has a Freiman-isomorphic copy in [-N^{C^{O(1)}}, N^{C^{O(1)}}] and, for any K > 1, a partition into sets of size >> (KC)^{-O(1)} N with doubling at most (CK)^{O(1)}, each in a generalized arithmetic progression of dimension << (KC)^{O(1)}, plus a remainder of size << (KC)^{-10} N.
• Theorem 2.2 (p. 3): For A a finite set of nonzero integers there is an F_4-isomorphic set B of nonzero integers with max_x sum_{b in B} (phi - 1/3)(bx) >> log log |B|, where phi is the indicator of (1/3, 2/3) on R/Z; this implies Theorem 1.2.