The Erdos-Turan problem in infinite groups

TL;DR

This paper solves the Erdos-Turan problem for infinite abelian groups by characterizing when such groups have perfect additive bases, revealing a complete classification based on the group's structure.

Contribution

It provides a complete classification of infinite abelian groups regarding the existence of perfect additive bases, including explicit constructions and limitations.

Findings

Groups not decomposable into a group of exponent 3 and order 2 have perfect additive bases.

Groups of the form of a group of exponent 3 plus order 2 lack perfect bases but have bases with at most two representations.

For exponent 2 groups, there exists a subset with bounded representations for non-zero elements.

Abstract

Let $G$ be an infinite abelian group with $|2G|=|G|$. We show that if $G$ is not the direct sum of a group of exponent 3 and the group of order 2, then $G$ possesses a perfect additive basis; that is, there is a subset $S\subseteq G$ such that every element of $G$ is uniquely representable as a sum of two elements of $S$. Moreover, if $G$ \emph{is} the direct sum of a group of exponent 3 and the group of order 2, then it does not have a perfect additive basis; however, in this case there is a subset $S\subseteq G$ such that every element of $G$ has at most two representations (distinct under permuting the summands) as a sum of two elements of $S$. This solves completely the Erdos-Turan problem for infinite groups. It is also shown that if $G$ is an abelian group of exponent 2, then there is a subset $S\subseteq G$ such that every element of $G$ has a representation as a sum of two elements of $S$, and the number of representations of non-zero elements is bounded by an absolute constant.