Box-counting dimension of separated digit sets (${\dim }_{\mathrm{M}}E=\log m/\log b$)

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 $\log m/\log b$.