Densities for sets of natural numbers vanishing on a given family

Rafał Filipów , Jacek Tryba


Abstract 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) [3] 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.
Author Rafał Filipów (FMPI / IM)
Rafał Filipów,,
- Institute of Mathematics
, Jacek Tryba (FMPI / IM)
Jacek Tryba,,
- Institute of Mathematics
Journal seriesJournal of Number Theory, ISSN 0022-314X, e-ISSN 1096-1658, (N/A 100 pkt)
Issue year2020
Publication size in sheets0.55
Keywords in Englishdensity of sets of integers, abstract upper density, submeasure, ideal of sets, Baire property, maximal almost disjoint family, ideal convergence
ASJC Classification2602 Algebra and Number Theory
Languageen angielski
Score (nominal)100
Score sourcejournalList
ScoreMinisterial score = 100.0, 10-03-2020, ArticleFromJournal
Publication indicators Scopus SNIP (Source Normalised Impact per Paper): 2018 = 1.079; WoS Impact Factor: 2018 = 0.684 (2) - 2018=0.721 (5)
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