The main result of this article is a characterization of the permutations θ : N→N that map a set with zero asymptotic density into a set with zero asymptotic density; a permutation has this property if and only if the lower asymptotic density of Cp tends to 1 as p→∞ where p is an arbitrary natural number and Cp = { l : θ−1 (l) ≤ lp } . We then show that a permutation has this property if and only if it maps statistically convergent sequences into statistically convergent sequences