= Box-counting dimension
{title2=$\underline{\dim}_{\mathrm M}E,\ \overline{\dim}_{\mathrm M}E$}
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= Minkowski dimension
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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 $\delta\downarrow0$. If these limits agree, their common value is its <Minkowski dimension>. A cover by $N_\delta(E)$ balls implies $|E_\delta|\lesssim_n\delta^nN_\delta(E)$ 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>.
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