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Box-counting content

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

Definition

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

B ( A ) := lim sup x N B ( A , x ) x D B and B ( A ) := lim inf x N B ( A , x ) x D B {\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 N B ( A , x ) {\displaystyle N_{B}(A,x)} is the maximum number of disjoint closed balls with centers a A {\displaystyle a\in A} and radii x 1 > 0 {\displaystyle x^{-1}>0} .

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

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

Examples

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

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

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

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

See also

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.
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