Box-counting dimension (${\underline{\dim }}_{\mathrm{M}}E,{\overline{\dim }}_{\mathrm{M}}E$)

For a nonempty bounded subset $E$ of Euclidean space, let $N_{\delta }(E)$ be the least number of radius-$\delta$ balls covering $E$. Its lower and upper box-counting dimensions are the lower and upper limits of $\log N_{\delta }(E)/\log (1/\delta )$ as $\delta \downarrow 0$. If these limits agree, their common value is its Minkowski dimension. A cover by $N_{\delta }(E)$ balls implies $|E_{\delta }|{\lesssim }_n{\delta }^n{N}_{\delta }(E)$ for the open $\delta$-neighborhood. Conversely, a maximal separated subset gives the reverse comparison up to fixed changes of scale. This connects small-scale covering counts with neighborhood Lebesgue measure.

Consider base-$b$ expansions using $m$ allowed digits with positive gaps between the corresponding first-level intervals. At depth $k$ there are $m^k$ cylinders of diameter comparable to $b^{-k}$, and separated cylinders cannot all be covered by fewer than a constant multiple of $m^k$ balls of that scale. Thus $N_{b^{-k}}(E)\asymp m^k$. Monotonicity between successive scales gives equal upper and lower Minkowski dimensions, both $\log m/\log b$.