Generalized average box dimensions of representative compact metric spaces
*Besma AyedCorresponding authorbesma.ayed@fsm.rnu.tnayed_besma@yahoo.frDepartment of MathematicsFaculty of Sciences of MonastirAlgerba, Number Theory and Nonlinear Analysis Laboratory, LR18ES15University of MonastirMonastir, 5000, TunisiaView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 May 2025
- Published Online:
- 18 May 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JIM-2402
- Pages:
- 1–20
Abstract
Keywords
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References
[1] R. Achour, J. Hattab, and B. Selmi, “New fractal dimensions of measures and decompositions of singularly continuous measures”, Fuzzy Sets and Systems, vol. 479, pp. 108859 (2024).
[2] R. Achour, Z. Li, B. Selmi, and T Wang, “A Multifractal Formalism for New General Fractal Measures”, Chaos Solitons & Fractals, vol. 181, pp. 114655 (2024).
[3] R. Achour, Z. Li, B. Selmi, and T. Wang, “General fractal dimensions of graphs of products and sums of continuous functions and their decompositions”, Journal of Mathematical Analysis and Applications, vol. 538, pp. 128400 (2024).
[4] R. Achour and B. Selmi, “General fractal dimensions of typical sets and measures”, Fuzzy Sets and Systems, vol. 490, pp. 109039 (2024).
[5] R. Achour and B. Selmi, “Some properties of new general fractal measures”, Monatshefte für Mathematik, vol. 204, pp. 659-678 (2024).
[6] R. Achour, Z. Li, and B. Selmi, “Variational principles for general fractal dimensions”, Results in Mathematics, vol. 79, pp. 261 (2024).
[7] R. Achour, S. Doria, B. Selmi, and Z. Li, “Generalized fractal dimensions and gauges for self-similar sets and their application in the assessment of coherent conditional previsions and in the calculation of the Sugeno integral”, Chaos, Solitons & Fractals, vol. 196, pp. 116374 (2025).
[8] D. Adam-Day, C. Ashcroft, L. Olsen, N. Pinzani, A. Rizzoli, and J. Rowe, “On the average box dimensions of graphs of typical continuous functions”, Acta Mathematica Hungarica, vol. 156, pp. 263-302 (2018).
[9] D. Allen, H. Edwards, S. Harper, and L. Olsen, “Average distances on self-similar sets and higher order average distances of self-similar measures”, Mathematische Zeitschrift, vol. 287, pp. 287-324 (2017).
[10] D. Cheng, Z. Li, and B. Selmi, “On the general fractal dimensions of hyperspace of compact sets”, Fuzzy Sets and Systems, vol. 488, pp. 108998 (2024).
[11] S. Doria and B. Selmi, “Conditional aggregation operators defined by the Choquet integral and the Sugeno integral with respect to general fractal measures”, Fuzzy Sets and Systems, vol. 477, pp. 108-811 (2024).
[12] K. J. Falconer, Fractal Geometry, John Wiley & Sons (1989).
[13] S. Havlin, SV. Buldyrev, AL. Goldberger, RN. Mantegna, SM. Ossadnik, CK. Peng, M. Simons, and HE. Stanley, “Fractals in biology and medicine”, Chaos, Solitons & Fractals, vol. 6, pp. 171-201 (1995).
[14] J.E. Hutchinson, “Fractals and Self Similarity”, Indiana University Mathematics Journal, vol. 30, no. 5, pp. 713-747 (1981).
[15] P. Gruber, “Dimension and Structure of Typical Compact Sets, Continua and Curves”, Monatshefte für Mathematik, vol. 108, pp. 149-164 (1989).
[16] G. H. Hardy, Divergent Series, London: The Clarendon Press (1949).
[17] M. Jacob, “Über die Äquivalenz der Cesàroschen und der Hölderschen Mittel für In tegrale bei gleicher reeller Ordnung k>0”, Mathematische Zeitschrift, vol. 26, pp. 672-682 (1927).
[18] Z. Li, B. Selmi, “On the Divergence of General Local and Fractal Dimensions of Typical Measures”, Real Anal. Exchange Advance, pp. 1-41 (2025). https://doi.org/10.14321/realanalexch.1745378977
[19] Z. Li, B. Selmi and H. Zyoudi, “A comprehensive approach to multifractal analysis”, Expositiones Mathematicae, vol. 43, pp. 125690 (2025).
[20] B. Mandelbrot, Fractals, Form, Chance, and Dimension, San Francisco: W. H. Freeman (1977).
[21] B. Mandelbrot,The fractal geometry of nature, San Francisco: W. H. Freeman (1983).
[22] B. Mandelbrot, “The Variation of Certain Speculative Prices”, The Journal of Business, vol. 36, no. 4, pp. 394-419 (1963).
[23] J. Myjak and R. Rudnicki, “Box and Packing Dimensions of Typical Compact Sets”, Monatshefte für Mathematik, vol. 131, pp. 223-226 (2000).
[24] L. Olsen, “On the Average Box Dimensions of Typical Compact Metric Spaces”, Kyungpook Mathematical Journal, vol. 64, pp. 633-647 (2024).
[25] L. Olsen, “On average Hewitt-Stromberg measures of typical compact metric spaces”, Mathematische Zeitschrift, vol. 293, pp. 1201-1225 (2019).
[26] L. Olsen, “On the average Lp-dimensions of typical measures belonging to the Gromov-Hausdorff-Prohoroff space”, Journal of Mathematical Analysis and Applications, vol. 469, pp. 916-934 (2019).
[27] L. Olsen, “Average box dimensions of typical compact sets”, Annales Academiæ Scientiarum Fennicæ Mathematica, vol. 44, pp. 141-165 (2019).
[28] L. Olsen and M. West, “Average frequencies of digits in infinite IFS’s and applications to continued fractions and Lüroth expansions”, Monatshefte für Mathematik, vol. 193, pp. 441-478 (2020).
[29] L. Olsen and A. Richardson, “Average distances between points in graph-directed self-similar fractals”, Mathematische Nachrichten, vol. 292, pp. 170-194 (2019).
[30] J. Oxtoby, Measure and category, A survey of the analogies between topological and measure spaces, New York, Berlin: Springer-Verlag (1971).
[31] P. Petersen, Riemannian Geometry, New York: Springer-Verlag (2006).
[32] J. Rouyer, “Generic properties of compact metric spaces”, Topology and its Applications, vol. 158, pp. 2140-2147 (2011).
[33] B. Selmi, “General multifractal dimensions of measures”, Fuzzy Sets and Systems, vol. 499, pp. 109177 (2025).
[34] B. Selmi, “Subsets of positive and finite Ψ_t-Hausdorff measures and applications”, The Journal of Geometric Analysis, vol. 34, no. 79 (2024).
[35] B. Selmi and H. Zyoudi, “Regarding the set-theoretic complexity of the general fractal dimensions and measures maps”, Analysis, vol. 45, no. 1, pp. 85-103 (2025).
[36] B. Selmi, “Average general fractal dimensions of typical compact metric spaces”, Fuzzy Sets and Systems, vol. 499, pp. 109192 (2025).
[37] B. Selmi, “Average Hewitt-Stromberg and box dimensions of typical compact metric spaces”, Quaestiones Mathematicæ, vol. 46, pp. 411-444 (2023).
[38] M. Shuminoska, E. Hadzieva, and E. Celakoska, “A Novel Web Application for Modeling 3D Fractals”, Egyptian Computer Science Journal, vol. 42, no. 1, pp. 43-56 (2018).
[39] D. Stroock, Essentials of Integration Theory for Analysis, Springer Verlag (2011).
[40] T. Vicsek, “Fractal models for diffusion controlled aggregation”, Journal of Physics A: Mathematical and General, vol. 16, pp. 647-652 (1983).
[41] T. Wang, B. Selmi, Z. Li, “General Hewitt-Stromberg measures: Properties and their role in multifractal formalism”, Contemp. Math., vol. 825, pp. 181-200 (2025).
[42] B. Yu, B. Selmi, and Y. Liang, “General fractal dimensions of graphs of continuous functions associated with the Katugampola fractional integral”, Fractals, vol. 33, pp. 2550040 (2025).
