Abstract
We address the following question: Is it true that, for every metric compactum \( X \) of box dimension \( \dim_{B}X=a\leq\infty \) and every two reals \( \alpha \) and \( \beta \) such that \( 0\leq\alpha\leq\beta\leq a \), there exists a closed subset in \( X \) whose lower box dimension is \( \alpha \) and whose upper box dimension is \( \beta \)? We give the positive answer for \( \alpha=0 \). In the general case, this result is final. We construct an example of a metric compactum whose box dimension is 1 but every nonempty proper closed subset of the compactum has lower box dimension 0.
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References
Pontryagin L. and Shnirelman L., “On one metric property of dimension,” Ann. Math., vol. 33, 156–162 (1932).
Pesin Ya.B., Dimension Theory in Dynamical Systems: Contemporary Views and Applications, The University of Chicago, Chicago and London (1997).
Graf S. and Luschgy H., Foundations of Quantization for Probability Distributions, Springer, Berlin (2000).
Ivanov A.V. and Fomkina O.V., “On the order of metric approximation of maximal linked systems and capacitarian dimensions,” Tr. Karelian Research Center of the Russian Academy of Sciences, vol. 7, 5–14 (2019).
Ivanov A.V., “On metric order in spaces of the form \( F(X) \),” Topology Appl., vol. 221, 107–113 (2017).
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The study was carried out under the State Task to the Institute of Applied Mathematical Research of the Karelian Scientific Center of the Russian Academy of Sciences.
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Translated from Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 3, pp. 540–545. https://doi.org/10.33048/smzh.2023.64.307
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Ivanov, A.V. On the Intermediate Values of the Box Dimensions. Sib Math J 64, 593–597 (2023). https://doi.org/10.1134/S0037446623030072
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DOI: https://doi.org/10.1134/S0037446623030072