Author(s): Dániel TÓTH

Title: SPECIÁLIS HALMAZOK MAXIMÁLIS ASZIMPTOTIKUS SŰRŰSÉGŰ RÉSZHALMAZAI

Source: F. Filip, Zs. Gódány, Š. Gubo, E. Korcsmáros, S. Tóbiás Kosár, T. Zsigmond (eds.): 16th International Conference of J. Selye University. Sections of the Faculty of Economics and Informatics. Conference Proceedings

ISBN: 978-80-8122-509-3

DOI: https://doi.org/10.36007/5093.2024.424

Publisher: J. Selye University, Komárno, Slovakia

PY, pages: 2024, 424-430

Published on-line: 2024

Language: hu

Abstract: The asymptotic density d, defined by lim_{n→∞} |A∩[1, n]|/n, is a classical tool for measuring the size of subsets of N. However, it is not defined for all subsets of N. In this article, we investigate two finitely additive measures that extend asymptotic density and are defined for all subsets of N. The density measures we consider, for a given subset A ⊆ N, assign the largest asymptotic density possessed by subsets of A, denoted by \underline{\underline d}}(A) = sup{d(B) | B ⊆ A, \underline{d}(B) = \overline{d}(B)}, and the smallest asymptotic density possessed by supersets of A, denoted by \overline{\overline d}}(A) = inf{d(C) | C ⊇ A, {\underline d}(C) = {\overline d} (C)}. We study the range of the density measures \underline{\underline d} , \underline d, \overline{\overline d}, and \overline d. Additionally, we construct sets with a specific form such that the values of \underline{\underline d} , \underline d, \overline{\overline d}, and \overline d can be assigned arbitrarily. From this result, it follows that there exist sets with arbitrarily large lower asymptotic density, strictly less than one, that contain no subset with positive asymptotic density.

Keywords: density measures, asymptotic density, Pólya density

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