On the $S$-Euclidean minimum of an ideal class
We show that the $S$-Euclidean minimum of an ideal class is a rational number, generalizing a result of Cerri. In the proof, we actually obtain a slight refinement of this and 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. In particular, we resolve a conjecture of Lenstra except when the $S$-units have rank one. The proof is self-contained but uses ideas from ergodic theory and topological dynamics, particularly those of Berend.