Densities on Dedekind domains, completions and Haar measure

TL;DR

This paper explores the relationship between density in rings of S-integers and Haar measure in their profinite completions, providing criteria for their equality, extending classical theorems, and analyzing prime elements.

Contribution

It introduces a general density definition, characterizes when density equals Haar measure, extends the Davenport-Erdős theorem, and analyzes prime element closures in Dedekind domains.

Findings

Density equals Haar measure under specific conditions.

Extended Davenport-Erdős theorem to all such domains.

Set of elements divisible by at most k primes has density zero.

Abstract

Let $D$ be the ring of $S$-integers in a global field and $\hat{D}$ its profinite completion. We discuss the relation between density in $D$ and the Haar measure of $\hat{D}$: in particular, we ask when the density of a subset $X$ of $D$ is equal to the Haar measure of its closure in $\hat{D}$. 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 $D$ 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 natural number $k$. Finally, we show that the closure of the set of prime elements of $D$ is the union of the group of units of $\hat{D}$ with a negligible part.