Generalized Whitney partitions

Michał Rams

Fundamenta Mathematicae (2000)

  • Volume: 166, Issue: 3, page 233-249
  • ISSN: 0016-2736

Abstract

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We prove that the upper Minkowski dimension of a compact set Λ is equal to the convergence exponent of any packing of the complement of Λ with polyhedra of size not smaller than a constant multiple of their distance from Λ.

How to cite

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Rams, Michał. "Generalized Whitney partitions." Fundamenta Mathematicae 166.3 (2000): 233-249. <http://eudml.org/doc/212479>.

@article{Rams2000,
abstract = {We prove that the upper Minkowski dimension of a compact set Λ is equal to the convergence exponent of any packing of the complement of Λ with polyhedra of size not smaller than a constant multiple of their distance from Λ.},
author = {Rams, Michał},
journal = {Fundamenta Mathematicae},
keywords = {Whitney partitions; Minkowski dimension; compact set; polyhedra},
language = {eng},
number = {3},
pages = {233-249},
title = {Generalized Whitney partitions},
url = {http://eudml.org/doc/212479},
volume = {166},
year = {2000},
}

TY - JOUR
AU - Rams, Michał
TI - Generalized Whitney partitions
JO - Fundamenta Mathematicae
PY - 2000
VL - 166
IS - 3
SP - 233
EP - 249
AB - We prove that the upper Minkowski dimension of a compact set Λ is equal to the convergence exponent of any packing of the complement of Λ with polyhedra of size not smaller than a constant multiple of their distance from Λ.
LA - eng
KW - Whitney partitions; Minkowski dimension; compact set; polyhedra
UR - http://eudml.org/doc/212479
ER -

References

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  1. [1] C. Bishop, Minkowski dimension and the Poincaré exponent, Michigan Math. J. 43 (1996), 231-246. Zbl0862.30042
  2. [2] C. Bishop, Geometric exponents and Kleinian groups, Invent. Math. 127 (1997), 33-50. Zbl0876.30044
  3. [3] L. Carleson, P. W. Jones and J. C. Yoccoz, Julia and John, Bol. Soc. Brasil. Mat. 25 (1994), 1-30. 
  4. [4] K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, Wiley, Chichester, 1990. 
  5. [5] O. Martio and M. Vuorinen, Whitney cubes, p-capacity and Minkowski content, Exposition. Math. 5 (1987), 17-40. Zbl0632.30023
  6. [6] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Univ. Press, Cambridge, 1995. Zbl0819.28004
  7. [7] P. J.Nicholls, The Ergodic Theory of Discrete Groups, Cambridge Univ. Press, Cambridge, 1989. Zbl0674.58001
  8. [8] C. Pommerenke, Boundary Behaviour of Conformal Maps, Springer, Heidelberg, 1992. 
  9. [9] M. Rams, Box dimension and self-intersecting Cantor sets, doctoral thesis, IM PAN, 1999 (in Polish). 
  10. [10] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1970. Zbl0207.13501
  11. [11] C. Tricot, Porous surfaces, Constr. Approx. 5 (1989), 117-136. 
  12. [12] C. Tricot, Curves and Fractal Dimension, Springer, Berlin, 1995. 
  13. [13] C. Tricot, Mesures et dimensions, doctoral thesis, Univ. Paris-Sud, 1983. 

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