A survey of complex dimensions, measurability, and the lattice/nonlattice dichotomy
The theory of complex dimensions of fractal strings developed by Lapidus and van Frankenhuijsen has proven to be a powerful tool for the study of Minkowski measurability of fractal subsets of the real line. In a very general setting, the Minkowski measurability of such sets is characterized by the s...
| Huvudupphovsmän: | , , , , |
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| Övriga upphovsmän: | |
| Materialtyp: | Artikel |
| Språk: | en_US |
| Publicerad: |
American Institute of Mathematical Sciences (AIMS)
2017
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| Länkar: | http://hdl.handle.net/1721.1/110078 https://orcid.org/0000-0002-7550-4049 |
| Sammanfattning: | The theory of complex dimensions of fractal strings developed by Lapidus and van Frankenhuijsen has proven to be a powerful tool for the study of Minkowski measurability of fractal subsets of the real line. In a very general setting, the Minkowski measurability of such sets is characterized by the structure of corresponding complex dimensions. Also, this tool is particularly effective in the setting of self-similar fractal subsets of R which have been shown to be Minkowski measurable if and only if they are nonlattice. This paper features a survey on the pertinent results of Lapidus and van Frankenhuijsen and a preliminary extension of the theory of complex dimensions to subsets of Euclidean space, with an emphasis on self-similar sets that satisfy various separation conditions. This extension is developed in the context of box-counting measurability, an analog of Minkowski measurability, which is shown to be characterized by complex dimensions under certain mild conditions. |
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