Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

6 Issues per year

IMPACT FACTOR 2017: 0.314
5-year IMPACT FACTOR: 0.462

CiteScore 2017: 0.46

SCImago Journal Rank (SJR) 2017: 0.339
Source Normalized Impact per Paper (SNIP) 2017: 0.845

Mathematical Citation Quotient (MCQ) 2017: 0.26

See all formats and pricing
More options …
Volume 68, Issue 4


Weighted uniform density ideals

Jacek Tryba
Published Online: 2018-08-06 | DOI: https://doi.org/10.1515/ms-2017-0139


Weighted uniform densities are a generalization of the uniform density, which is also known as the Banach density. In this paper, we introduce the concept of weighted uniform density ideals and consider the topological complexity of these ideals as well as when they have certain analytical properties related to the ideal convergence of sequences and series. Furthermore, we prove some inequalities between different upper and lower weighted uniform densities and give the answer to the problem concerning the Darboux property of these densities.

MSC 2010: Primary 11B05; Secondary 40A35; 40A05

Keywords: uniform density; Banach density; weighted uniform density; Darboux property; ideal; filter; P-ideal; ideal convergence; 𝓘-convergent series


  • [1]

    Alexander, R.: Density and multiplicative structure of sets of integers, Acta Arith. 12 (1966/1967), 321–332.Google Scholar

  • [2]

    Barbarski, P.—Filipów, R.—Mrożek, N.—Szuca, P.: Uniform density u and 𝓘u-convergence on a big set, Math. Commun. 16 (2011), 125–130.Google Scholar

  • [3]

    Červeňanský, J.—Šalát, T.—Toma, V.: Remarks on statistical and I-convergence of series, Math. Bohem. 130 (2005), 177–184.Google Scholar

  • [4]

    Farah, I.: Analytic quotients: theory of liftings for quotients over analytic ideals on the integers, Mem. Amer. Math. Soc. 148 (2000), 177 pp.Google Scholar

  • [5]

    Farah, I.: How many Boolean algebras 𝓟(ℕ)/𝓘 are there?, Illinois J. Math. 46 (2002), 999–1033.Google Scholar

  • [6]

    Filipów, R.—Mrożek, N.—Recław, I.—Szuca, P.: Ideal convergence of bounded sequences, J. Symbolic Logic 72 (2007), 501–512.Web of ScienceCrossrefGoogle Scholar

  • [7]

    Gáliková, Z.—László, B.—Šalát, T.: Remarks on uniform density of sets of integers, Acta Acad. Paedagog. Agriensis Sect. Mat. (N.S.) 29 (2002), 3–13.Google Scholar

  • [8]

    Giuliano Antonini, R.—Grekos, G.: Weighted uniform densities, J. Théor. Nombres Bordeaux 19 (2007), 191–204.CrossrefGoogle Scholar

  • [9]

    Głąb, S.—Olczyk, M.: Convergence of series on large set of indices, Math. Slovaca 65 (2015), 1095–1106.Web of ScienceGoogle Scholar

  • [10]

    Grekos, G.—Mišík, L.—Ziman, M.: I-convergence and (T) property, manuscript, 2004.Google Scholar

  • [11]

    Just, W.—Krawczyk, A.: On certain Boolean algebras 𝓟(ω)/I, Trans. Amer. Math. Soc. 285 (1984), 411–429.Google Scholar

  • [12]

    Ki, H.—Linton, T.: Normal numbers and subsets ofwith given densities, Fund. Math. 144 (1994), 163–179.Google Scholar

  • [13]

    Mačaj, M.—Mišík, L.—Šalát, T.—Tomanová, J.: On a class of densities of sets of positive integers, Acta Math. Univ. Comenian. (N.S.) 72 (2003), 213–221.Google Scholar

  • [14]

    Mačaj, M.—Sleziak, M.—Toma, V.: On weighted uniform density, Unif. Distrib. Theory 3 (2008), 101–127.Google Scholar

  • [15]

    Tripathy, B. C.: On statistically convergent series, Punjab Univ. J. Math. (Lahore) 32 (1999), 1–7.Google Scholar

  • [16]

    Tryba, J.: Subseries of 𝓘-convergent series, Lith. Math. J. 58 (2018), 104–112.CrossrefWeb of ScienceGoogle Scholar

About the article

Received: 2016-05-10

Accepted: 2017-06-06

Published Online: 2018-08-06

Published in Print: 2018-08-28

Communicated by Ján Borsík

Citation Information: Mathematica Slovaca, Volume 68, Issue 4, Pages 717–726, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0139.

Export Citation

© 2018 Mathematical Institute Slovak Academy of Sciences.Get Permission

Comments (0)

Please log in or register to comment.
Log in