Density sets of sets of positive integers
type: Item
instance of: scholarly article
title: Density sets of sets of positive integers
zbMATH Open document ID: 1203.11013
author: Georges Grekos
author: János T. Tóth
author: Ladislav Mišík
published in: Journal of Number Theory
publication date: 2010-06-18T00:00:00Z
review text: A nonnegative function \(f\) on the set of positive integers induces a concept of upper density by \[ \overline d(A) = \limsup \Bigl( \sum _{a\leq n, a\in A} f(a) \Bigr) \Bigm/ \sum _{k=1}^n f(k), \] and a lower density analogously. Given a set \(A\), its density set defined by \[ S(A) = \{ ( \overline d(X), \underline d(X) ): X\subset A\} \subset [0,1]^2 . \] For a set which fails to have a density this contains more information than just the upper and lower densities. The authors describe this set for a wide class of functions, which include familiar examples like the asymptotic and logarithmic density.
reviewed by: Imre Z. Ruzsa
Mathematics Subject Classification ID: 11B05
zbMATH DE Number: 5723356
zbMATH Keywords: density
zbMATH Keywords: asymptotic density
zbMATH Keywords: weighted density
zbMATH Keywords: density set
MaRDI profile type: Publication
full work available at URL: j.jnt.2009.12.007
OpenAlex ID: W2081963739
cites work: Density and multiplicative structure of sets of integers
cites work: Répartition des densites des sous-suites d'une suite d'entiers
cites work: On densities and gaps
cites work: Q5291006
cites work: Q3346413
DOI: 10.1016/J.JNT.2009.12.007
description: scientific article; zbMATH DE number 5723356