On orders in quadratic number fields with unusual sets of distances

Abstract

Let $\mathcal O$ be an order in an algebraic number field and suppose that the set of distances $\varDelta (\mathcal O)$ of $\mathcal O$ is nonempty (equivalently, $\mathcal O$ is not half-factorial). If $\mathcal O$ is seminormal (in particular, if $\mat