Sum-avoiding sets in groups

Van Vu,Terence Tao

doi:10.19086/da.887

Abstract

Let $A$ be a finite subset of an arbitrary additive group $G$, and let $\phi(A)$ denote the cardinality of the largest subset $B$ in $A$ that is sum-avoiding in $A$ (that is to say, $b_1+b_2 \not \in A$ for all distinct $b_1,b_2 \in B$). The question of controlling the size of $A$ in terms of $\phi(A)$ in the case when $G$ was torsion-free was posed by Erd\H{o}s and Moser. When $G$ has torsion, $A$ can be arbitrarily large for fixed $\phi(A)$ due to the presence of subgroups. Nevertheless, we provide a qualitative answer to an analogue of the Erd\H{o}s-Moser problem in this setting, by establishing a structure theorem, which roughly speaking asserts that $A$ is either efficiently covered by $\phi(A)$ finite subgroups of $G$, or by fewer than $\phi(A)$ finite subgroups of $G$ together with a residual set of bounded cardinality. In order to avoid a large number of nested inductive arguments, our proof uses the language of nonstandard analysis. We also answer negatively a question of Erd\H{o}s regarding large subsets $A$ of finite additive groups $G$ with $\phi(A)$ bounded, but give a positive result when $|G|$ is not divisible by small primes.

Highlights

Let G = (G, +) be an additive group
The main result of this paper is to establish a partial converse to this observation, covering A efficiently by up to k groups, or up to k − 1 groups and a set of bounded cardinality: Theorem 1.2 (Small φ implies covering by groups)
Let us see how Theorem 2.4 implies Theorem 1.2; the converse implication is not needed here and is left to the interested reader. This is a routine application of the transfer principle in nonstandard analysis, but for the convenience of the reader we provide a self-contained argument

Summary

Introduction

When the group G contains a lot of torsion, removing a large subgroup H from A can leave one with a residual set with no good additive structure, and in particular with no bounds whatsoever on φ (A\H). It is easy to see that φ (A) is at most 2, but upon removing the large finite group H from A one is left with an arbitrary subset of x + H, and in particular φ (A\H) can be arbitrarily large The problem in this example is that the group H is the “incorrect” group to try to remove from A; one should instead remove the larger group H := H + {0, x}, which contains H as an index two subgroup. We are able to sharpen the classification in Theorem 1.2 when G is a finite group whose order is not divisible by small primes; see Theorem 1.5 below

A question of Erdos

Nonstandard analysis formulation

Main argument

Findings

Groups of order not divisible by small primes