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- W4287670009 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-ErdH{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." @default.
- W4287670009 created "2022-07-25" @default.
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- W4287670009 date "2020-09-09" @default.
- W4287670009 modified "2023-09-27" @default.
- W4287670009 title "Densities on Dedekind domains, completions and Haar measure" @default.
- W4287670009 doi "https://doi.org/10.48550/arxiv.2009.04229" @default.
- W4287670009 hasPublicationYear "2020" @default.
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