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 $\dim _{B}I$ 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 ${\mathcal {B}}(I)=1/2$.

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

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. arXiv:1510.06467. doi:10.3934/dcdss.2017011.