Research Article

A note on the density of k-free polynomial sets, Haar measure and global fields

Published in: Quaestiones Mathematicae
Volume 45 , issue 9 , 2022 , pages: 1373–1397

DOI: 10.2989/16073606.2021.1945701

Author(s): Luca Demangos , China , Ignazio Longhi , India

Keywords: Primary: 11R45 , Secondary: 11B05 , 11N25 , 11C08 , 11N32 , Primary: 11R45 , Secondary: 11B05 , 11N25 , 11C08 , 11N32

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Abstract

In this work we investigate the general relation between the density of a subset of the ring of integers D of a general global field and the Haar measure of its closure in the profinite completion . We then study a specific family of sets, the preimages of k-free elements (for any given k ∈ ℕ\{0, 1}) via one variable polynomial maps, showing that under some hypotheses their asymptotic density always exists and it is precisely the Haar measure of the closure in of their set.

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