Box-counting content

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

Definition

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

[math]\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}} }[/math]

where [math]\displaystyle{ N_B(A, x) }[/math] is the maximum number of disjoint closed balls with centers [math]\displaystyle{ a\in A }[/math] and radii [math]\displaystyle{ x^{-1} \gt 0 }[/math].

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

If [math]\displaystyle{ 0 \lt \mathcal{B}_*(A) \lt \mathcal{B}^*(A) \lt \infty }[/math], then [math]\displaystyle{ A }[/math] is said to be box-counting measurable.

Examples

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

[math]\displaystyle{ \dim_BI=\dim_MI=1. }[/math]

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

By contrast, [math]\displaystyle{ I }[/math] is Minkowski measurable with [math]\displaystyle{ \mathcal{M}(I) = 1 }[/math].

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.