Density of sets of natural numbers and the Levy group

Abstract

Let $\N$ denote the set of positive integers. The asymptotic density of the set $A \subseteq \N$ is $d(A) = \lim_{n\to\infty} |A\cap [1,n]|/n$, if this limit exists. Let $ \mathcal{AD}$ denote the set of all sets of positive integers that have asymptotic density, and let $S_{\N}$ denote the set of all permutations of the positive integers \N. The group $\mathcal{L}^{\sharp}$ consists of all permutations $f \in S_{\N}$ such that $A \in \mathcal{AD}$ if and only if $f(A) \in \mathcal{AD}$, and the group $\mathcal{L}^{\ast}$ consists of all permutations $f \in \mathcal{L}^{\sharp}$ such that $d(f(A)) = d(A)$ for all $A \in \mathcal{AD}$. Let $f:\N \to \N $ be a one-to-one function such that $d(f(\N))=1$ and, if $A \in \mathcal{AD}$, then $f(A) \in \mathcal{AD}$. It is proved that $f$ must also preserve density, that is, $d(f(A)) = d(A)$ for all $A \in \mathcal{AD}$. Thus, the groups $\mathcal{L}^{\sharp}$ and $\mathcal{L}^{\ast}$ coincide.