FRACTAL DIMENSIONS OF SETS DEFINED BY DIGIT RESTRICTIONS IN â„ 2

Lipeng Wang and Wenxia Li ()
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Lipeng Wang: School of Mathematical Sciences, East China Normal University, 500 Dongchuan Rd., Shanghai 200241 P. R. China
Wenxia Li: School of Mathematical Sciences, East China Normal University, 500 Dongchuan Rd., Shanghai 200241 P. R. China

FRACTALS (fractals), 2023, vol. 31, issue 07, 1-24

Abstract: We introduce a class of sets defined by digit restrictions in â„ 2 and study its fractal dimensions. Let ES,D be a set defined by digit restrictions in â„ 2. We obtain the Hausdorff and lower box dimensions of ES,D. Under some condition, we gain the packing and upper box dimensions of ES,D. We get the Assouad dimension of ES,D and show that it is 2 if and only if ES,D contains arbitrarily large arithmetic patches. Under some conditions, we study the upper spectrum, quasi-Assouad dimension and Assouad spectrum of ES,D. Finally, we give an intermediate value property of fractal dimensions of the class of sets.

Keywords: Sets Defined by Digit Restrictions in â„ 2; Assouad Dimension; Arithmetic Patch; Upper Spectrum; Quasi-Assouad Dimension; Assouad Spectrum (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1142/S0218348X23500743

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