Box-counting content

In mathematics, the box-counting content is an analog of Minkowski content.

Definition
Let $$A$$ be a bounded subset of $$m$$-dimensional Euclidean space $$\mathbb{R}^m$$ such that the box-counting dimension $$D_B$$ exists. The upper and lower box-counting contents of $$A$$ are defined by



\mathcal{B}^*(A) := \limsup_{x \rightarrow \infty} \frac{N_B(A, x)}{x^{D_B}}\quad\quad \text{and} \quad\quad \mathcal{B}_*(A) := \liminf_{x \rightarrow \infty} \frac{N_B(A, x)}{x^{D_B}} $$

where $$N_B(A, x)$$ is the maximum number of disjoint closed balls with centers $$a\in A$$ and radii $$x^{-1} > 0$$.

If $$\mathcal{B}^*(A) = \mathcal{B}_*(A)$$, then the common value, denoted $$\mathcal{B}(A)$$, is called the box-counting content of $$A$$.

If $$0 < \mathcal{B}_*(A) < \mathcal{B}^*(A) < \infty$$, then $$A$$ is said to be box-counting measurable.

Examples
Let $$I=[0,1]$$ denote the unit interval. Note that the box-counting dimension $$\dim_BI$$ and the Minkowski dimension $$\dim_MI$$ coincide with a common value of 1; i.e.


 * $$\dim_BI=\dim_MI=1.$$

Now observe that $$N_B(I, x) = \lfloor x/2\rfloor + 1$$, where $$\lfloor y \rfloor$$ denotes the integer part of $$y$$. Hence $$I$$ is box-counting measurable with $$\mathcal{B}(I) = 1/2$$.

By contrast, $$I$$ is Minkowski measurable with $$\mathcal{M}(I) = 1$$.