Abstract
In this note we present and comment three equivalent definitions of the so called uniform or Banach density of a set of positive integers.
Highlights
IntroductionThe concept of density of an infinite set of positive integers appeared to be a basic tool to obtain an idea of the magnitude and of the structure of subsets of positive integers
L’accès aux articles de la revue « Annales mathématiques Blaise Pascal », implique l’accord avec les conditions générales d’utilisation
In this note we present and comment three equivalent definitions of the so called uniform or Banach density of a set of positive integers
Summary
The concept of density of an infinite set of positive integers appeared to be a basic tool to obtain an idea of the magnitude and of the structure of subsets of positive integers. Lower) asymptotic density of a set A of positive integers, denoted by d(A) The aim of this note is to present and comment three equivalent definitions of the so called uniform or Banach density of a set of positive integers. Given a set A ⊆ N, its upper and lower Banach densities, denoted by b(A) and b(A) (see definitions in sections 1 and 3, respectively), satisfy the relation. The principal characteristic of the uniform density is that it is more sensitive to local density in any interval, not necessarily initial, than the asymptotic density. + n}, having rare but sufficiently long blocks of consecutive integers, has asymptotic density zero while its upper uniform density equals to 1 and so its uniform density does not exist.
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