Densities on Dedekind domains, completions and Haar measure

Resumen:

The notion of density is one of the classical benchmarks to understand the size of a set of integer numbers, a fundamental tool to study famous problems like the twin primes conjecture or the Artin conjecture on primitive roots. One of its main qualities is that it offers an effective alternative to the notion of probability, which is not available with desirable properties on Z. Occasionally the lack of such a notion may lead to substantial complications in using this technique.

We propose an approach based on embedding Z (or a more general class of Dedekind domains) into the profinite completion, where the natural notion of Haar measure is available. We explore conditions under which such a measure may coincide with the density of the initial set, by proposing a generalized notion of density and applying it to famous results in the area in order to test the potential of this approach.

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