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Box-counting content

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

Definition

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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

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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

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References

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  • 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.