On densities and gaps
type: Item
instance of: scholarly article
title: On densities and gaps
zbMATH Open document ID: 0622.10044
DOI: 10.1016/0022-314X(87)90074-6
author: Georges Grekos
author: Bodo Volkmann
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: The authors investigate certain relations between the asymptotic densities of subsets of a set \(A=\{m_ 1,m_ 2,...\}\) of positive integers and the lengths of gaps in A by means of the ''analytical'' properties of the two-dimensional set \(S(A)=\{(\underline d(B),\bar d(B))\in {\mathbb{R}}^ 2\); \(B\subseteq A\}\), where ḏ(B) and \(\bar d(B)\) denote the lower and upper asymptotic density of B. A special role here is played by the function \(f_ S\), whose graph is the lower part of the boundary of S(A). So for instance, the right derivatives \(D^+_ f(0)\) of f at the origin is not less than \(\lambda (A)=\limsup_{j\to \infty}m_{j+1}/m_ j\). Conversely, given a set \(S\subseteq {\mathbb{R}}^ 2\) satisfying certain conditions and a real number \(\lambda\) such that \(1\leq \lambda \leq D^+_ f(0)\), where \(f=f_ S\) then there exists a set A of positive integers such that \(S(A)=S\) and \(\lambda (A)=\lambda\). They also give a characterization of \(D^+_ f(0)\) of the function f corresponding to a given set A in terms of the lengths of intervals in which the set A possesses ''few'' elements, etc.
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11K38
zbMATH DE Number: 4008512
zbMATH Keywords: density set
zbMATH Keywords: asymptotic density
zbMATH Keywords: gap density
reviewed by: Štefan Porubský
MaRDI profile type: Publication
cites work: Répartition des densites des sous-suites d'une suite d'entiers
cites work: Q5848286
full work available at URL: 0022-314x(87)90074-6
OpenAlex ID: W1986352595
description: scientific article; zbMATH DE number 4008512