2020/09/09 by L. Demangos, Demangos, Luca, Ignazio Longhi +1
Mathematics · #Advanced Topology and Set Theory #Limits and Structures in Graph Theory #History and Theory of Mathematics
paper · pdf · doi:10.48550/arxiv.2009.04229
Let D be the ring of S-integers in a global field and D its profinite completion. We discuss the relation between density in D and the Haar measure of D: in particular, we ask when the density of a subset X of D is equal to the Haar measure of its closure in 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 D with a negligible part.