Small sets without unique products in torsion-free groups

Abstract

In this paper, we report on a computation demonstrating that if [Formula: see text] and [Formula: see text] are nonempty subsets of a torsion-free group such that [Formula: see text] has no unique product, then [Formula: see text]. Moreover, this bound is sharp, as there are examples where [Formula: see text], and in fact [Formula: see text] for at least two such examples. More generally, when [Formula: see text] is small, we find lower bounds on [Formula: see text]. One consequence of this work is that any counterexample to Kaplansky’s zero-divisor conjecture must be quite complicated, if it exists at all. Another advance is that we give new examples of torsion-free groups that are not unique product groups.