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On the Range of the Quantization Dimension of Probability Measures on a Metric Compactum

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Abstract

The quantization dimension of a probability measure on a metric compactum \( X \) does not exceed the box dimension of the support of the measure. We prove the following intermediate value theorem for the upper quantization dimension: If \( X \) is a metric compact space whose upper box dimension is equal to \( a\leq\infty \) then for every real \( b \) such that \( 0\leq b\leq a \) there exists a probability measure on \( X \) whose support is \( X \) and whose upper quantization dimension is \( b \).

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Funding

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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Correspondence to A. V. Ivanov.

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Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 5, pp. 1074–1081. https://doi.org/10.33048/smzh.2022.63.509

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Ivanov, A.V. On the Range of the Quantization Dimension of Probability Measures on a Metric Compactum. Sib Math J 63, 903–908 (2022). https://doi.org/10.1134/S0037446622050093

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  • DOI: https://doi.org/10.1134/S0037446622050093

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