TY - JOUR

T1 - Densities on Dedekind domains, completions and Haar measure

AU - Demangos, Luca

AU - Longhi, Ignazio

N1 - Publisher Copyright:
© 2024, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2024/2

Y1 - 2024/2

N2 - 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–Erdős 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.

AB - 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–Erdős 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.

UR - http://www.scopus.com/inward/record.url?scp=85181926084&partnerID=8YFLogxK

U2 - 10.1007/s00209-023-03415-2

DO - 10.1007/s00209-023-03415-2

M3 - Article

AN - SCOPUS:85181926084

SN - 0025-5874

VL - 306

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 2

M1 - 21

ER -