Density of Sets of Natural Numbers and the Lévy Group

Let N denote the set of positive integers. The asymptotic density of the set A ⊆ N is d(A) = limn→∞ |A ∩ [1, n]|/n, if this limit exists. Let AD denote the set of all sets of positive integers that have asymptotic density, and let SN denote the set of all permutations of the positive integers N. The group L consists of all permutations f ∈ SN such that A ∈ AD if and only if f(A) ∈ AD, and the group L∗ consists of all permutations f ∈ L such that d(f(A)) = d(A) for all A ∈ AD. Let f : N → N be a one-to-one function such that d(f(N)) = 1 and, if A ∈ AD, then f(A) ∈ AD. It is proved that f must also preserve density, that is, d(f(A)) = d(A) for all A ∈ AD. Thus, the groups L and L∗ coincide. 1. Asymptotic density and permutations Let A be a set of positive integers, and let A(n) = ∑ a∈A 1≤a≤n 1 denote the counting function of the set A. The lower asymptotic density of A is dL(A) = lim inf n→∞ A(n) n . The upper asymptotic density of A is dU (A) = lim sup n→∞ A(n) n . The set A has asymptotic density d(A) if the limit d(A) = lim n→∞ A(n) n exists. The set A has an asymptotic density if and only if dL(A) = dU (A). We denote by AD the set of all sets of positive integers that have asymptotic density, that is, AD = {A ⊆ N : dL(A) = dU (A)}. Let SN denote the group of all permutations of the positive integers N. For any set A ⊆ N and permutation g ∈ SN, we let g(A) = {g(a) : a ∈ A}. 2000 Mathematics Subject Classification. Primary 11B05, 11B13, 11B75.