Permutations that preserved asymptotically null sets and statistical convergence

Abstract

A permutation maps sets with asymptotic density zero to sets with asymptotic density zero if and only if the upper asymptotic density of { l : θ ^{−1} (l) > lp } tends to  0   as p → ∞ where  p is a natural number. A permutation has this property if and only if it maps statistically convergent sequences to statistically convergent sequences.