On the S-Euclidean minimum of an ideal class

Abstract

We show that the S-Euclidean minimum of an ideal class is a rational number, generalizing a result of Cerri. We also give some corollaries which explain the relationship of our results with Lenstra's notion of a norm-Euclidean ideal class and the conjecture of Barnes and Swinnerton-Dyer on quadratic forms. The proof is self-contained but uses ideas from ergodic theory and topological dynamics, particularly those of Berend.