Densities on Dedekind domains, completions and Haar measure

Abstract

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 k ∈ N.Finally, we show that the group of units of D is contained in the closure of the set of irreducible elements of D.

Local and global densities for Weierstrass models of elliptic curves

“…Assuming that v(a 1 ) ≥ 2, so a ∈ W(2, 2, 3, 3, 5), we have ∆ ≡ a 4 3 (mod 2 13 ), so n ≥ 12, with n = 12 ⇐⇒ v(a 3 ) = 3. Assuming further that v(a 3 ) ≥ 4, so a ∈ W(2, 2, 4, 3, 5), we have ∆ ≡ 2 4 (2 2 a 4 1 + 2 6 a 2 2 + a 2 6 ) ≡ 2 14 (a 4 1,2 + a 2 2,2 + a 2 6,5 ) (mod 2 15 ), so n ≥ 14, with n = 14 ⇐⇒ a 6,5 ̸ ≡ a 1,2 + a 2,2 (mod 2). Assuming that a 6,5 ≡ a 1,2 + a 2,2 (mod 2), we find that ∆ ≡ 2 15 (mod 2 16 ), so that n = 15.…”

Coset topologies on $\mathbb{Z}$ and arithmetic applications

“…For X ⊆ Z, let XT be the closure of X in ẐT ; in the case T = T F we shall write simply X. As we are going to discuss briefly in subsection 3.3 (and, hopefully, more extensively in future work -see also [9]), determining X appears quite meaningful for number theory. We don't know if the same is true for other péijí topologies.…”

“…This is nicely illustrated by the following theorem, which is one of the key inspirational ideas of our work. It is probably well-known to experts, but we are not aware of it appearing in the literature before (apart from a quick comment in [9,Remark 3.3]). A generalization to rings of S-integers of global fields can be found in […”

“…where * is an easy consequence of (28) (see [9,Example 4.3] for details). Thus, Furstenberg's argument is based on the fact that P has empty interior, while Euler's consists in computing its measure.…”

“…The comparison between density and Haar measure is a leading theme of [9], in a setting where both take the value 0 for P (which is regarded as a subset of the much bigger set Z). Theorem 3.23 suggests that the same approach can be applied to X ⊆ P, by taking (31) for the density and µ Ẑ * ( X ∩ Ẑ * ) for the measure.…”