Density of summable subsequences of a sequence and its applications

  • 1 School of Mathematics, Jilin University, 130012, Changchun, P. R. China
  • 2 School of Science, Nanjing University of Posts and Telecommunications, 210023, Nanjing, P. R. China
Bingzhe Hou, Yue Xin and Aihua Zhang

Abstract

Let x = {xn}n=1 be a sequence of positive numbers, and 𝓙x be the collection of all subsets A ⊆ ℕ such that kAxk < +∞. The aim of this article is to study how large the summable subsequence could be. We define the upper density of summable subsequences of x as the supremum of the upper asymptotic densities over 𝓙x, SUD in brief, and we denote it by D*(x). Similarly, the lower density of summable subsequences of x is defined as the supremum of the lower asymptotic densities over 𝓙x, SLD in brief, and we denote it by D*(x). We study the properties of SUD and SLD, and also give some examples. One of our main results is that the SUD of a non-increasing sequence of positive numbers tending to zero is either 0 or 1. Furthermore, we obtain that for a non-increasing sequence, D*(x) = 1 if and only if lim infknxn=0, which is an analogue of Cauchy condensation test. In particular, we prove that the SUD of the sequence of the reciprocals of all prime numbers is 1 and its SLD is 0. Moreover, we apply the results in this topic to improve some results for distributionally chaotic linear operators.

  • [1]

    Bayart, F.—Ruzsa, I.: Difference sets and frequently hypercyclic weighted shifts, Ergodic Theory Dynam. Systems 35(3) (2015), 691–709.

    • Crossref
    • Export Citation
  • [2]

    Bermúdez, T.—Bonilla, A.—Martínez-Giménez, F.–Peris, A.: Li-Yorke and distributionally chaotic operators, J. Math. Anal. Appl. 373(1) (2011), 83–93.

    • Crossref
    • Export Citation
  • [3]

    Bernardes Jr., N. C.—Bonilla, A.—Müller, V.—Peris, A.: Distributional chaos for linear operators, J. Funct. Anal. 265(9) (2013), 2143–2163.

    • Crossref
    • Export Citation
  • [4]

    Bernardes Jr. N. C., —Bonilla, A.—Peris, A.—Wu, X.: Distributional chaos for operators on Banach spaces, J. Math. Anal. Appl. 459(2) (2018), 797–821.

    • Crossref
    • Export Citation
  • [5]

    Chand, H.: On some generalizations of Cauchy’s condensation and integral tests, Amer. Math. Monthly 46(6) (1939), 338–341.

    • Crossref
    • Export Citation
  • [6]

    Di Nasso, M.—Jin, R.: Abstract densities and ideals of sets, Acta Arith. 185(4) (2018), 301–313.

    • Crossref
    • Export Citation
  • [7]

    Farah, I.: Analytic quotients: theory of liftings for quotients over analytic ideals on the integers, Mem. Amer. Math. Soc. 148(48) (2000).

  • [8]

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

  • [9]

    Grosse-Erdmann, K.: Hypercyclic and chaotic weighted shifts, Stud. Math. 139 (2000), 47–68.

    • Crossref
    • Export Citation
  • [10]

    Hou, B.—Tian, G.—Shi, L.: Some dynamical properties for linear operators, Illinois J. Math. 53(3) (2009), 857–864.

    • Crossref
    • Export Citation
  • [11]

    Hou, B.—Cui, P.—Cao, Y.: Chaos for Cowen-Douglas operators, Proc. Amer. Math. Soc. 138 (2010), 929–936.

    • Crossref
    • Export Citation
  • [12]

    Hou, B.—Tian, G.—Zhu, S.: Approximation of chaotic operators, J. Operator Theory 67(2) (2012), 469–493.

  • [13]

    Krzyś, J.: A theorem of Ollvier and its generalizations Prace matem. 2 (1956), 159–164 (in Polish).

  • [14]

    Kanovei, V.: Borel Equivalence Relations: Structure and Classification. University Lecture Series, vol. 44, American Mathematical Society, Providence, RI, 2008.

  • [15]

    Leonetti, P.—Tringali, S.: Upper and lower densities have the strong Darboux property, J. Number Theory 174 (2017), 445–455.

    • Crossref
    • Export Citation
  • [16]

    Leonetti, P.—Tringali, S.: On the notion of upper and lower density, Proc. Edinb. Math. Soc. (2019), https://doi.org/10.1017/S0013091519000208.

  • [17]

    Luo, L.—Hou, B.: Some remarks on distributional chaos for bounded linear operators, Turkish J. Math. 39(2) (2015), 251–258.

    • Crossref
    • Export Citation
  • [18]

    Martínez-Giménez, F.—Oprocha, P.—Peris, A.: Distributional chaos for backward shifts, J. Math. Anal. Appl. 351(2) (2009), 607–615.

    • Crossref
    • Export Citation
  • [19]

    Martínez-Giménez, F.—Oprocha, P.—Peris, A.: Distributional chaos for operators with full scrambled sets, Math. Z. 274 (2013), 603–612.

    • Crossref
    • Export Citation
  • [20]

    Moritz, R.: On the extended form of Cauchy’s condensation test for the convergence of infinite series, Bull. Amer. Math. Soc. 44(6) (1938), 441–442.

    • Crossref
    • Export Citation
  • [21]

    Moser, L. On the series ∑ 1/p, Amer. Math. Monthly 65 (1958), 104–105.

    • Crossref
    • Export Citation
  • [22]

    Powell, B. J.—Šalát, T.: Convergence of subseries of the harmonic series and asymptotic densities of sets of positive integers, Publications de l’Institut Mathematique, Nouvelle serie tome 50(64) (1991), 60–70.

  • [23]

    Šalát, T.: On subseries, Math. Z. 85 (1964), 209–225.

  • [24]

    Schweizer, B.—Smítal, J.: Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc. 344(2) (1994), 737–754.

    • Crossref
    • Export Citation
  • [25]

    Tryba, J.: Weighted uniform density ideals, Math. Slovaca 68(4) (2018), 717–726.

    • Crossref
    • Export Citation
  • [26]

    Wu, X.—Zhu, P.—Lu, T.: Uniform distributional chaos for weighted shift operators, Appl. Math. Lett. 26(1) (2013), 130–133.

    • Crossref
    • Export Citation
  • [27]

    Yin, Z.—He, S.—Huang, Y.: On Li-Yorke and distributionally chaotic direct sum operators, Topology Appl. 239 (2018), 35–45.

    • Crossref
    • Export Citation
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