I'm trying to understand the definition of Minkowski dimension given by Wikipedia here.

In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a set $S$ in a Euclidean space $\mathbb{R}^n$, or more generally in a metric space $(X, d)$.

It's defined as

$$\textrm{dim}(S)=\lim\limits_{\epsilon\to 0}\frac{\log N(\epsilon)}{\log(1/\epsilon)}$$ where $N(\epsilon)$ is the minimum number of balls of radius $\epsilon$ required to cover $S$.

If the above limit does not exist, one may still take the limit superior and limit inferior, which respectively define the upper box dimension and lower box dimension.

My question(s): can $S$ be anything or do we have to restrict ourselves to bounded (or even compact?) subsets of $\mathbb{R}^n$? If we don't, $N(\epsilon)$ can be infinite. In this case I guess the definition would lead us to consider that the Minkowski dimension is infinite, right? For example, $\textrm{dim}(\mathbb{R})=\infty$?

In the more general case of a metric space $(X,d)$, I have the same kind of questions. Do we have to consider special conditions on $(X,d)$ (like bounded, compact, locally compact) or on the subset $S$ of $X$?

By the way, if you have some good references about Minkowski dimension, Hausdorff measure and dimension, and related subjects...


1 Answer 1


You are correct - the set $S$ must be bounded for this definition to be applicable. Even if the set is bounded, there are issues with this definition. For example, it's easy to see that $\dim(S)=1$ for $S=[0,1]\cap\mathbb Q$, even though it's a countable set. Focusing on compact sets doesn't really fix the problem. If $S=\{0,1,1/2,1/3,\ldots,1/n,\ldots\}$ then $\dim(S)=1/2$, though this is a bit more work to prove.

These examples illustrate that box-counting dimension is not $\sigma$-stable. That is, we can have $$\dim\left(\bigcup_{n=1}^{\infty} E_n\right) > \sup_n \dim(E_n),$$ which is certainly an undesirable property in any notion of dimension. There is a standard way to modify the definition, namely define $$\dim_M(S) = \inf\left\{\sup_n \dim(S_n): S=\bigcup_{n=1}^{\infty} S_n\right\}.$$ That is, we consider all ways to decompose the set $S$ into countably many pieces and return the smallest possible value of the largest dimension from all decompositions. The result is not only $\sigma$-stable (alleviating the problems above) but is also applicable to unbounded sets.

Box-counting directly is still very nice for at least three reasons:

  • It is very easy to compute relative to measure theoretic definitions, like the Hausdorff dimension,
  • It provides an easy upper bound for most other dimensions,
  • It can be estimated experimentally for many natural objects.

The modified box-counting dimension, while nicer from a theoretical standpoint, loses these advantages.


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