Demangos, L and Longhi, I (2024) Densities on Dedekind domains, completions and Haar measure. In: Mathematische Zeitschrift, 306 (2).
|
PDF
Mat_zei_306_2_2024.pdf - Published Version Download (835kB) | Preview |
Abstract
Let D be the ring of S-integers in a global field and D its profinite completion. Given X� Dn, we consider its closure X � D n and ask what can be learned from X about the �size� of X. In particular, we ask when the density of X is equal to the Haar measure of X. We provide a general definition of density which encompasses the most commonly used ones. Using it we give a necessary and sufficient condition for the equality between density and measure which subsumes a criterion due to Poonen and Stoll. We also show how Ekedahl�s sieve fits into our setting and find conditions ensuring that X can be written as a product of local closures. In another direction, we extend the Davenport�Erd�s theorem to every D as above and offer a new interpretation of it as a �density=measure� result. Our point of view also provides a simple proof that in any D the set of elements divisible by at most k distinct primes has density 0 for any k� N. Finally, we show that the closure of the set of prime elements of D is the union of the group of units of D with a negligible part. © 2024, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
| Item Type: | Journal Article |
|---|---|
| Publication: | Mathematische Zeitschrift |
| Publisher: | Springer Science and Business Media Deutschland GmbH |
| Additional Information: | The copyright for this article belongs to Author. |
| Department/Centre: | Division of Physical & Mathematical Sciences > Mathematics |
| Date Deposited: | 29 Feb 2024 11:24 |
| Last Modified: | 29 Feb 2024 11:24 |
| URI: | https://eprints.iisc.ac.in/id/eprint/83921 |
Actions (login required)
 |
View Item |
