Density of sets of natural numbers and the Levy group

Details Density of sets of natural numbers and the Levy group Density of sets of natural numbers and the Levy group

2006-04-04

arxiv.org/abs/math/0604055v1

Mathematics

Number Theory

6 pages

Scientific paper

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.

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