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词条 Box-counting content
释义

  1. Definition

  2. Examples

  3. See also

  4. References

{{expert needed|date=March 2017|reason=Possible duplicate article, or part of a series. }}

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

Definition

Let be a bounded subset of -dimensional Euclidean space such that the box-counting dimension exists.

The upper and lower box-counting contents of are defined by

where is the maximum number of disjoint closed balls with centers

and radii .

If , then the common value, denoted , is called the box-counting content of .

If , then is said to be box-counting measurable.

Examples

Let denote the unit interval.

Note that the box-counting dimension and the Minkowski dimension coincide with a common value of 1; i.e.

Now observe that , where denotes the integer part of . Hence is box-counting measurable with .

By contrast, is Minkowski measurable with .

See also

  • Box counting

References

  • {{cite journal|last1=Dettmers|first1=Kristin|last2=Giza|first2=Robert|last3=Morales|first3=Rafael|last4=Rock|first4=John A.|last5=Knox|first5=Christina|title=A survey of complex dimensions, measurability, and the lattice/nonlattice dichotomy|journal=Discrete and Continuous Dynamical Systems - Series S|date=January 2017|volume=10|issue=2|pages=213–240|doi=10.3934/dcdss.2017011|arxiv=1510.06467}}
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