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

${{ℬ}}^{*}(A):=\underset{x\to \mathrm{\infty }}{\mathrm{lim\; sup}}\frac{N_B(A,x)}{x^{D_B}}\text{and}{{ℬ}}_{*}(A):=\underset{x\to \mathrm{\infty }}{\mathrm{lim\; inf}}\frac{N_B(A,x)}{x^{D_B}}$

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

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

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

Examples

Let $I=[0,1]$ denote the unit interval. Note that the box-counting dimension ${\dim }_{B}I$ and the Minkowski dimension ${\dim }_{M}I$ coincide with a common value of 1; i.e.

${\dim }_{B}I={\dim }_{M}I=1.$

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 ${ℬ}(I)=1/2$.

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

See also

• Box counting

References

• Dettmers, Kristin; Giza, Robert; Morales, Rafael; Rock, John A.; Knox, Christina (January 2017). "A survey of complex dimensions, measurability, and the lattice/nonlattice dichotomy". Discrete and Continuous Dynamical Systems - Series S 10 (2): 213–240. doi:10.3934/dcdss.2017011.