Densities on Dedekind domains, completions and Haar measure
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-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 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.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2020
- DOI:
- 10.48550/arXiv.2009.04229
- arXiv:
- arXiv:2009.04229
- Bibcode:
- 2020arXiv200904229D
- Keywords:
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- Mathematics - Number Theory
- E-Print:
- 41 pages, no figures. Final version, accepted in Mathematische Zeitschrift