The Erdős-turán Problem in Infinite Groups

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 ⊆ G such that every element of G is uniquely representable as a sum of two elements of S. Moreover, if G 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 ⊆ 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 Erdős-Turán problem for infinite groups. It is also shown that if G is an abelian group of exponent 2, then there is a subset S ⊆ 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.