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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa171-2-2,
author = {Kevin J. McGown},
title = {On the S-Euclidean minimum of an ideal class},
journal = {Acta Arithmetica},
volume = {168},
year = {2015},
pages = {125-144},
zbl = {06497305},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa171-2-2}
}
Kevin J. McGown. On the S-Euclidean minimum of an ideal class. Acta Arithmetica, Tome 168 (2015) pp. 125-144. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa171-2-2/