Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-19T12:03:31.167Z Has data issue: false hasContentIssue false
  • English
  • Français

A new proof for the residual set dimension of the apollonian packing

Published online by Cambridge University Press:  24 October 2008

Claude Tricot
Claude Tricot
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C. V6T 1Y4, Canada
Get access Rights & Permissions [Opens in a new window]

Abstract

Let (Dn) be the apollonian packing of a curvilinear triangle T, ρn the radius of Dn, E = T—U Dn the residual set, dim (E) its Hausdorff dimension. In this paper we give a new proof of the equality dim proved by Boyd [2]. Our technique is to construct a sequence of regular triangles covering E, and suitable measures μkcarried by E which allow us to apply a density theorem.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 3 , November 1984 , pp. 413 - 423
Copyright
Copyright © Cambridge Philosophical Society 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Boyd, D. W.. Osculatory packings by spheres. Canad. Math. Bull. 13 (1970), 59–64.CrossRefGoogle Scholar
[2]Boyd, D. W.. The residual set dimension of the apollonian packing. Mathematika 20 (1973), 170–174.CrossRefGoogle Scholar
[3]Hirst, K. E.. The apollonian packing of circles. J. Lond. Math. Soc. 42 (1967), 281–291.CrossRefGoogle Scholar
[4]Kasner, E. and Stjpnick, F.. The Apollonian packing of circles. Proc. Nat. Acad. Sci. U.S.A. 29 (1943), 378–384.CrossRefGoogle ScholarPubMed
[5]Melzak, Z. A.. Infinite packings of disks. Canad. J. Math. 18 (1966), 838–852.CrossRefGoogle Scholar
[6]Rogers, C. A. and Taylor, S. J.. Functions continuous and singular with respect to a Hausdorff measure. Mathematika 8 (1961), 1–31.CrossRefGoogle Scholar
[7]Tricot, C.. Metric properties of compact sets of measure 0 in the plane, included in Mesures et dimensions. These de doctorat (Paris, 1983).Google Scholar
[8]Wilker, J. B.. Open disk packings of a disk. Canad. Math. Bull. 10 (1967), 395–415.CrossRefGoogle Scholar