GENERALIZED CANTOR-INTEGERS AND INTERVAL DENSITY OF HOMOGENEOUS CANTOR SETS

Jin Chen and Xin-Yu Wang ()
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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

FRACTALS (fractals), 2023, vol. 31, issue 09, 1-9

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].

Keywords: Generalized Cantor-Integers; Self-Similar Measure; Interval Density (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1142/S0218348X23500998

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