On the S-Euclidean minimum of an ideal class

TL;DR

This paper proves that the S-Euclidean minimum of an ideal class is always rational, extending previous results and connecting to conjectures in quadratic forms, using ergodic theory and topological dynamics.

Contribution

It generalizes Cerri's result by establishing the rationality of the S-Euclidean minimum for ideal classes and explores its implications in number theory.

Findings

The S-Euclidean minimum is rational for ideal classes.

Connections between Euclidean minima and quadratic form conjectures.

Application of ergodic theory to algebraic number theory.

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.