3 research outputs found
Densities on Dedekind domains, completions and Haar measure
Let be the ring of -integers in a global field and its
profinite completion. We discuss the relation between density in and the
Haar measure of : in particular, we ask when the density of a subset
of is equal to the Haar measure of its closure in . In order
to have a precise statement, we give a general definition of density which
encompasses the most commonly used ones. Using it we provide a necessary and
sufficient condition for the equality between density and measure which
subsumes a criterion due to Poonen and Stoll. In another direction, we extend
the Davenport-Erd\H{o}s theorem to every 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 the set of elements divisible by at
most distinct primes has density 0 for any natural number . Finally, we
show that the closure of the set of prime elements of is the union of the
group of units of with a negligible part.Comment: 39 pages, no figures. Main changes from version 3: most examples of
densities have been removed and will be part of another paper; some questions
have been added in the final section. Version 5: Remark 6.13 was based on a
wrong computation and has been remove