Box-counting content

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

Definition

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

${\displaystyle {\mathcal {B}}^{*}(A):=\limsup _{x\rightarrow \infty }{\frac {N_{B}(A,x)}{x^{D_{B}}}}\quad \quad {\text{and}}\quad \quad {\mathcal {B}}_{*}(A):=\liminf _{x\rightarrow \infty }{\frac {N_{B}(A,x)}{x^{D_{B}}}}}$

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

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

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

Examples

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

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

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

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

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.