Density of sets of natural numbers and the Lévy group

✍ Scribed by Melvyn B. Nathanson; Rohit Parikh

• Publisher: Elsevier Science
• Year: 2007
• Tongue: English
• Weight: 112 KB
• Volume: 124
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2006.09.001

✦ Synopsis

Let N denote the set of positive integers. The asymptotic density of the set

/n, if this limit exists. Let 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 L consists of all permutations f ∈ S N 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.