---
tag: F7G7CJY143
kind: definition
name: Minkowski–Bouligand dimension
msc: [28A80, 28A75]
defines: WYRKJ412FT
uses: [09QSZVVM26, 0KW3PN547K, 0T3RZQXDYD, 0BXNDZFQME, SSNFTHZ6YJ, HS2NTCMZ4T]
sources:
  - url: https://en.wikipedia.org/wiki/Minkowski%E2%80%93Bouligand_dimension
---

## Definition

Let $A$ be a nonempty [[bounded set|09QSZVVM26]] in $\R^n$.
For $\varepsilon > 0$, let $N(A, \varepsilon)$ be the smallest number of sets of [[diameter|0KW3PN547K]] at most $\varepsilon$ that cover $A$.
The *upper* and *lower box-counting dimensions* of $A$ are
$$\overline{\dim}_B A = \limsup_{\varepsilon \to 0^+} \frac{\log N(A, \varepsilon)}{\log (1/\varepsilon)}, \qquad \underline{\dim}_B A = \liminf_{\varepsilon \to 0^+} \frac{\log N(A, \varepsilon)}{\log (1/\varepsilon)}.$$
If they are equal, the common value $\dim_B A$ is the *Minkowski–Bouligand dimension*, or box-counting dimension, of $A$.

## Remarks

One gets the same values if $N(A, \varepsilon)$ counts the cubes of an $\varepsilon$-mesh grid that meet $A$, or the largest number of disjoint balls of radius $\varepsilon$ centred in $A$.
For bounded $A$, $\overline{\dim}_B A = \inf \set{ s : \mathcal{M}^{*s}(A) = 0 }$, where $\mathcal{M}^{*s}$ is the upper [[Minkowski content|HS2NTCMZ4T]].
Always $\dim_H A \le \underline{\dim}_B A \le \overline{\dim}_B A$, where $\dim_H$ is the [[Hausdorff dimension|SSNFTHZ6YJ]].

Example: let $C$ be the middle-thirds [[Cantor set|0BXNDZFQME]].
It is covered by $2^k$ intervals of length $3^{-k}$, and each set of diameter at most $3^{-k}$ meets at most two of them.
So $2^{k-1} \le N(C, 3^{-k}) \le 2^k$, and the ratio tends to $\log 2 / \log 3$, so $\dim_B C = \log 2 / \log 3$.

The box dimension does not ignore countable sets.
The set $\Q \cap [0, 1]$ is dense, so $N(\Q \cap [0, 1], \varepsilon) = N([0, 1], \varepsilon)$, and its box dimension is $1$.
Its Hausdorff dimension is $0$.

## Explanations

### L1 human

Count how many small sets of size $\varepsilon$ are needed to cover the set.
If the count grows like $\varepsilon^{-d}$ as $\varepsilon$ shrinks, the box dimension is $d$.