Sum-free subsets of finite abelian groups of type III

TL;DR

This paper classifies the largest sum-free subsets in finite abelian groups of type III, characterizes near-largest sum-free subsets, and provides formulas and bounds for their counts and symmetries, extending previous results.

Contribution

It offers a complete classification of maximum sum-free subsets in type III abelian groups and analyzes their structure and enumeration, extending prior work.

Findings

Classification of largest sum-free subsets in type III groups

Formula for the number of automorphism orbits on these subsets

Improved upper bounds for the number of sum-free subsets in such groups

Abstract

A finite abelian group $G$ of cardinality $n$ is said to be of type III if every prime divisor of $n$ is congruent to 1 modulo 3. We obtain a classification theorem for sum-free subsets of largest possible cardinality in a finite abelian group $G$ of type III. This theorem, when taken together with known results, gives a complete characterisation of sum-free subsets of the largest cardinality in any finite abelian group $G$. We supplement this result with a theorem on the structure of sum-free subsets of cardinality "close" to the largest possible in a type III abelian group $G$. We then give two applications of these results. Our first application allows us to write down a formula for the number of orbits under the natural action of ${\rm Aut}(G)$ on the set of sum-free subsets of $G$ of the largest cardinality when $G$ is of the form $({\mathbf{Z}}/m{\mathbf{Z}})^r$, with all prime divisors of $m$ congruent to 1 modulo 3, thereby extending a result of Rhemtulla and Street. Our second application provides an upper bound for the number of sum-free subsets of $G$. For finite abelian groups $G$ of type III and with {\em a given exponent} this bound is substantially better than that implied by the bound for the number of sum-free subsets in an arbitrary finite abelian group, due to Green and Ruzsa.