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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.

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