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.
@article{bwmeta1.element.bwnjournal-article-fmv166i3p233bwm, author = {Micha\l\ Rams}, title = {Generalized Whitney partitions}, journal = {Fundamenta Mathematicae}, volume = {163}, year = {2000}, pages = {233-249}, zbl = {0962.05016}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv166i3p233bwm} }
Rams, Michał. Generalized Whitney partitions. Fundamenta Mathematicae, Tome 163 (2000) pp. 233-249. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv166i3p233bwm/
[00000] [1] C. Bishop, Minkowski dimension and the Poincaré exponent, Michigan Math. J. 43 (1996), 231-246. | Zbl 0862.30042
[00001] [2] C. Bishop, Geometric exponents and Kleinian groups, Invent. Math. 127 (1997), 33-50. | Zbl 0876.30044
[00002] [3] L. Carleson, P. W. Jones and J. C. Yoccoz, Julia and John, Bol. Soc. Brasil. Mat. 25 (1994), 1-30.
[00003] [4] K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, Wiley, Chichester, 1990.
[00004] [5] O. Martio and M. Vuorinen, Whitney cubes, p-capacity and Minkowski content, Exposition. Math. 5 (1987), 17-40. | Zbl 0632.30023
[00005] [6] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Univ. Press, Cambridge, 1995. | Zbl 0819.28004
[00006] [7] P. J.Nicholls, The Ergodic Theory of Discrete Groups, Cambridge Univ. Press, Cambridge, 1989. | Zbl 0674.58001
[00007] [8] C. Pommerenke, Boundary Behaviour of Conformal Maps, Springer, Heidelberg, 1992.
[00008] [9] M. Rams, Box dimension and self-intersecting Cantor sets, doctoral thesis, IM PAN, 1999 (in Polish).
[00009] [10] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1970. | Zbl 0207.13501
[00010] [11] C. Tricot, Porous surfaces, Constr. Approx. 5 (1989), 117-136.
[00011] [12] C. Tricot, Curves and Fractal Dimension, Springer, Berlin, 1995.
[00012] [13] C. Tricot, Mesures et dimensions, doctoral thesis, Univ. Paris-Sud, 1983.