ArticleOriginal scientific text

Title

Generalized Whitney partitions

Authors 1

Affiliations

  1. Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warszawa, Poland

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.

Bibliography

  1. C. Bishop, Minkowski dimension and the Poincaré exponent, Michigan Math. J. 43 (1996), 231-246.
  2. C. Bishop, Geometric exponents and Kleinian groups, Invent. Math. 127 (1997), 33-50.
  3. L. Carleson, P. W. Jones and J. C. Yoccoz, Julia and John, Bol. Soc. Brasil. Mat. 25 (1994), 1-30.
  4. K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, Wiley, Chichester, 1990.
  5. O. Martio and M. Vuorinen, Whitney cubes, p-capacity and Minkowski content, Exposition. Math. 5 (1987), 17-40.
  6. P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Univ. Press, Cambridge, 1995.
  7. P. J.Nicholls, The Ergodic Theory of Discrete Groups, Cambridge Univ. Press, Cambridge, 1989.
  8. C. Pommerenke, Boundary Behaviour of Conformal Maps, Springer, Heidelberg, 1992.
  9. M. Rams, Box dimension and self-intersecting Cantor sets, doctoral thesis, IM PAN, 1999 (in Polish).
  10. E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1970.
  11. C. Tricot, Porous surfaces, Constr. Approx. 5 (1989), 117-136.
  12. C. Tricot, Curves and Fractal Dimension, Springer, Berlin, 1995.
  13. C. Tricot, Mesures et dimensions, doctoral thesis, Univ. Paris-Sud, 1983.
Pages:
233-249
Main language of publication
English
Received
1998-06-04
Accepted
1999-07-06
Published
2000
Exact and natural sciences