Geodesic
On the box dimension of subsets of a metric compact space
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 83 (2023), pp. 24-30

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The question of possible values of the lower capacity dimension $\underline{\mathrm{dim}}_B$ of subsets of the metric compact set $X$ is considered. The concept of dimension $f\underline{\mathrm{dim}}_BX$ is introduced, which characterizes the asymptotics of the lower capacity dimension of closed $\varepsilon$-neighborhoods of finite subsets of the compact set $X$ for $\varepsilon\to0$. For a wide class of metric compact sets, the dimension $f\underline{\mathrm{dim}}_BX$ is the same as $\underline{\mathrm{dim}}_BX$. The following theorem is proved: for any non-negative number $r$ there exists a closed subset $Z_r\subset X$ such that $\underline{\mathrm{dim}}_BZ_r=r$.
Keywords: metric compact space, capacitarian dimension, intermediate value theorem for the capacitarian dimension.
Mots-clés : quantization dimension 
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     author = {A. V. Ivanov},
     title = {On the box dimension of subsets of a metric compact space},
     journal = {Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika},
     pages = {24--30},
     publisher = {mathdoc},
     number = {83},
     year = {2023},
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     url = {http://geodesic.mathdoc.fr/item/VTGU_2023_83_a2/}
}
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A. V. Ivanov. On the box dimension of subsets of a metric compact space. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 83 (2023), pp. 24-30. http://geodesic.mathdoc.fr/item/VTGU_2023_83_a2/