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

Benjamin Bedert

TL;DR

This work resolves a long-standing question in additive combinatorics by proving that every finite set of integers A contains a sum-free subset A′ of size at least |A′| ≥ |A|/3 + c log log |A| for some absolute c>0. The authors develop a Fourier-analytic framework around the sum-free test function F_A, establish inverse theorems linking small L^1-norms to additive dimension, and construct a dense Freiman model via small-dimension structure. A novel non-Archimedean test-function method, together with a detailed analysis modulo small primes, yields strong lower bounds on the relevant L^1 norms and reveals a global structure: A embeds into a short interval under Freiman isomorphism, with large-energy behavior on all large subsets. The combination of these ingredients leads to the 99% structure theorem, providing a comprehensive picture of how sum-free subsets arise in dense, structured environments and advancing the Erdős program toward sharper asymptotics.

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.

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

TL;DR

This work resolves a long-standing question in additive combinatorics by proving that every finite set of integers A contains a sum-free subset A′ of size at least |A′| ≥ |A|/3 + c log log |A| for some absolute c>0. The authors develop a Fourier-analytic framework around the sum-free test function F_A, establish inverse theorems linking small L^1-norms to additive dimension, and construct a dense Freiman model via small-dimension structure. A novel non-Archimedean test-function method, together with a detailed analysis modulo small primes, yields strong lower bounds on the relevant L^1 norms and reveals a global structure: A embeds into a short interval under Freiman isomorphism, with large-energy behavior on all large subsets. The combination of these ingredients leads to the 99% structure theorem, providing a comprehensive picture of how sum-free subsets arise in dense, structured environments and advancing the Erdős program toward sharper asymptotics.

Abstract

A set is said to be \emph{sum-free} if there are no with . We show that there exists a constant such that any set of integers contains a sum-free subset of size . This answers a longstanding problem in additive combinatorics, originally due to Erdős.

Paper Structure

This paper contains 10 sections, 43 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Setup and paper overview
  3. Notation and prerequisites
  4. The Fourier series of $\sum_{a\in A}\left(\phi-1/3\right)(ax)$
  5. The structure of sets with small $L^1$-norm
  6. Sets with small dimension have a 'dense' model
  7. The distribution of $A$ modulo small primes
  8. Non-Archimedean test functions
  9. The global structure of sets with $S(A)\leqslant N/3+C$
  10. The McGehee-Pigno-Smith test function

Key Result

Theorem 1.2

There exists some constant $c>0$ such that for all finite sets $A\subset\mathbf{Z}$ we have $S(A)\geqslant \frac{|A|}{3}+c\log \log |A|$. In particular, $S(N)\geqslant \frac{N}{3}+c\log\log N$.

Theorems & Definitions (79)

  • Theorem 1.2
  • Theorem 1.3: 99% Structure Theorem
  • Definition 2.1
  • Theorem 2.2
  • Theorem 3.1: Littlewood's $L^1$ conjecture
  • Corollary 3.2
  • Lemma 3.3: M-P-S basic construction of test functions
  • proof
  • Definition 3.4
  • Theorem 3.5: Rudin's inequality
  • ...and 69 more