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 (Figure presented.). 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 (Figure presented.) of their set.
Original language | English |
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Pages (from-to) | 1373-1397 |
Number of pages | 25 |
Journal | Quaestiones Mathematicae |
Volume | 45 |
Issue number | 9 |
DOIs | |
Publication status | Published - 2022 |
Keywords
- Densities
- Haar measure
- global fields
- k-free values of polynomials
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Demangos, L., & Longhi, I. (2022). A note on the density of k-free polynomial sets, Haar measure and global fields. Quaestiones Mathematicae, 45(9), 1373-1397. https://doi.org/10.2989/16073606.2021.1945701