On the box dimension of subsets of a metric compact space

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 $r<f\underline{\mathrm{dim}}_BX$ there exists a closed subset $Z_r\subset X$ such that $\underline{\mathrm{dim}}_BZ_r=r$.