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Generalized Cantor-Integers And Interval Density Of Homogeneous Cantor Sets

Author

Listed:
  • JIN CHEN

    (College of Informatics, Huazhong Agricultural University, Wuhan 430070, P. R. China)

  • XIN-YU WANG

    (College of Informatics, Huazhong Agricultural University, Wuhan 430070, P. R. China)

Abstract

In this paper, we study the generalized Cantor-integers {Cn}n≥1 with the base conversion function f : {0,…,m}→ [0,p] being strictly increasing and satisfying f(0) = 0 and f(m) = p. We show that the sequence {Cn nα }n≥1 with α =logm+1(p+1) is dense in the closed interval with the endpoints being its inferior and superior, respectively. Moreover, every homogeneous Cantor set ℭ satisfying open set condition can be induced by some generalized Cantor-integers, we get the exact point which attains the maximal interval density of the form [0,x] with respect to the self-similar probability measure supported on ℭ. This result partially confirms a conjecture of E. Ayer and R. S. Strichartz [Exact Hausdorff measure and intervals of maximum density for Cantor sets, Trans. Am. Math. Soc. 351(9) (1999) 3725–3741].

Suggested Citation

  • Jin Chen & Xin-Yu Wang, 2023. "Generalized Cantor-Integers And Interval Density Of Homogeneous Cantor Sets," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 31(09), pages 1-9.
    • Handle: RePEc:wsi:fracta:v:31:y:2023:i:09:n:s0218348x23500998
    DOI: 10.1142/S0218348X23500998
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