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The residual set dimension of the Apollonian packing

Published online by Cambridge University Press:  26 February 2010

David W. Boyd
David W. Boyd
Affiliation:
Department of Mathematics, The University of British Columbia, Vancouver, Canada
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In this paper we show that, for the Apollonian or osculatory packing C0 of a curvilinear triangle T, the dimension d(C0, T) of the residual set is equal to the exponent of the packing e(Co, T) = S. Since we have [5, 6] exhibited constructible sequences λ(K) and μ(K) such that λ(K) < S < μ(K), and μ(K)–λ(K) → 0 as κ → 0, we have thus effectively determined d(C0, T). In practical terms it is thus now known that 1·300197 < d(C0, T) < 1·314534.

Type
Research Article
Information
Mathematika , Volume 20 , Issue 2 , December 1973 , pp. 170 - 174
Copyright
Copyright © University College London 1973

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