Densities for sets of natural numbers vanishing on a given family
Rafał Filipów , Jacek Tryba
AbstractAbstract upper densities are monotone and subadditive functions from the power set of positive integers into the unit real interval that generalize the upper densities used in number theory, including the upper asymptotic density, the upper Banach density, and the upper logarithmic density. At the open problem session of the Workshop “Densities and their application”, held at St. Étienne in July 2013, G. Grekos asked a question whether there is a “nice” abstract upper density, whose the family of null sets is precisely a given ideal of subsets of , where “nice” would mean the properties of the familiar densities consider in number theory. M. Di Nasso, R. Jin (2018)  showed that the answer is positive for the summable ideals (for instance, the family of finite sets and the family of sequences whose series of reciprocals converge) when “nice” density means translation invariant and rich density (i.e. density which is onto the unit interval). In this paper we extend their result to all ideals with the Baire property.
|Journal series||Journal of Number Theory, ISSN 0022-314X, e-ISSN 1096-1658, (N/A 100 pkt)|
|Publication size in sheets||0.55|
|Keywords in English||density of sets of integers, abstract upper density, submeasure, ideal of sets, Baire property, maximal almost disjoint family, ideal convergence|
|Score||= 100.0, 10-03-2020, ArticleFromJournal|
|Publication indicators||: 2018 = 1.079; : 2018 = 0.684 (2) - 2018=0.721 (5)|
* presented citation count is obtained through Internet information analysis and it is close to the number calculated by the Publish or Perish system.