A NOTE ON THE DENSITY OF K-FREE POLYNOMIAL SETS, HAAR MEASURE AND GLOBAL FIELDS

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) over cap. We then study a specific family of sets, the preimages of k-free elements (for any given k is an element of 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) over cap of their set.