[Paper Review] Counting sum-free sets in Abelian groups

Noga Alon, József Balogh+2 more|arXiv (Cornell University)|Jan 31, 2012

Limits and Structures in Graph Theory24 references5 citations

TL;DR

This paper establishes a sparse analogue of Green and Ruzsa's theorem on sum-free sets in finite Abelian groups, proving that for groups of order divisible by a prime $ q \equiv 2 \pmod{3} $, almost all sum-free subsets of size $ m \geq C(q)\sqrt{n\log n} $ are contained in a maximum-size sum-free subset. The authors develop a general framework using 3-uniform hypergraphs and spectral methods to derive structural results in the sparse regime from dense stability results, yielding precise asymptotic counts and typical structure for such sets.

ABSTRACT

In this paper we study sum-free sets of order $m$ in finite Abelian groups. We prove a general theorem on 3-uniform hypergraphs, which allows us to deduce structural results in the sparse setting from stability results in the dense setting. As a consequence, we determine the typical structure and asymptotic number of sum-free sets of order $m$ in Abelian groups $G$ whose order is divisible by a prime $q$ with $q \equiv 2 \pmod 3$, for every $m \ge C(q) \sqrt{n \log n}$, thus extending and refining a theorem of Green and Ruzsa. In particular, we prove that almost all sum-free subsets of size $m$ are contained in a maximum-size sum-free subset of $G$. We also give a completely self-contained proof of this statement for Abelian groups of even order, which uses spectral methods and a new bound on the number of independent sets of size $m$ in an $(n,d,λ)$-graph.

Motivation & Objective

• To establish a sparse analogue of Green and Ruzsa's structural result on sum-free sets in finite Abelian groups.
• To determine the typical structure and asymptotic number of sum-free sets of size $ m $ in Abelian groups $ G $ whose order is divisible by a prime $ q \equiv 2 \pmod{3} $.
• To develop a general method for deriving sparse structural results from dense stability results in additive combinatorics.
• To prove that almost all sum-free subsets of size $ m $ are contained in a maximum-size sum-free subset for $ m \geq C(q)\sqrt{n\log n} $.

Proposed method

• Develops a general theorem on 3-uniform hypergraphs to link dense stability results to sparse structural results.
• Applies spectral methods to bound the number of independent sets of size $ m $ in $ (n,d,\lambda) $-graphs.
• Uses character theory and eigenvalue bounds to analyze Cayley graphs associated with sum-free sets.
• Employs Chernoff's inequality and union bounds to control concentration of eigenvalues over random subsets.
• Leverages the structure of characters of finite Abelian groups as eigenvectors of Cayley graph adjacency matrices.
• Combines hypergraph regularity with spectral techniques to prove that large independent sets in sparse graphs are contained in large independent sets of the whole graph.

Experimental results

Research questions

• RQ1 What is the typical structure of sum-free subsets of size $ m $ in finite Abelian groups when $ m $ is large but sublinear?
• RQ2 How many sum-free subsets of size $ m $ exist in Abelian groups of Type I (i.e., with a prime divisor $ q \equiv 2 \pmod{3} $)?
• RQ3 Are almost all sum-free sets of size $ m $ contained in a maximum-size sum-free subset for such groups?
• RQ4 Can stability results in the dense setting be used to derive structural results in the sparse setting for sum-free sets?

Key findings

• For Abelian groups $ G $ of order $ n $ divisible by a prime $ q \equiv 2 \pmod{3} $, almost all sum-free subsets of size $ m \geq C(q)\sqrt{n\log n} $ are contained in a maximum-size sum-free subset of $ G $.
• The asymptotic number of sum-free sets of size $ m $ in such groups is $ \binom{n/2}{m} $ up to a subexponential factor, with the dominant contribution coming from sets inside maximum-size sum-free sets.
• A new spectral bound on the number of independent sets of size $ m $ in $ (n,d,\lambda) $-graphs is established, which is central to the proof.
• The paper provides a self-contained proof for groups of even order using spectral methods and the new independent set bound.
• The framework allows transferring structural stability results from dense to sparse settings in additive combinatorics via hypergraph and spectral techniques.