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

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 D. We then study a specific family of sets, the preimages of k-free elements (for any given k є 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 D of their set.