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Epsilon entropy and the packing of balls in Euclidean space

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

John Hawkes
John Hawkes
Affiliation:
Department of Mathematics, University College of Swansea, Singleton Park, Swansea. SA2 8PP.
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Summary

Let be a sequence of mutually disjoint open balls, with centres xj and corresponding radii aj, each contained in the closed unit ball in d-dimensional euclidean space, ℝd. Further we suppose, for simplicity, that the balls Bj are indexed so that aj≥aj+1. The set

obtained by removing, from the balls {Bj} is called the residual set. We say that the balls {Bj} constitute a packing of provided that λ(ℛ)=0, where λ denotes the d-dimensional Lebesgue measure. Thus it follows that henceforth denoted by c(d), whilst the packing restraint ensures that Larman [11] has noted that, under these circumstances, one also has .

MSC classification

Secondary: 28A80: Fractals
Type
Research Article
Information
Mathematika , Volume 43 , Issue 1 , June 1996 , pp. 23 - 31
Copyright
Copyright © University College London 1996

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References

1.Besicovitch, A. S. and Taylor, S. J.. On the complementary intervals of a linear closed set of zero Lebesgue measure. J. London Math. Soc., 20 (1954), 449–459.CrossRefGoogle Scholar
2.Boyd, D. W.. The residual set dimension of the Apollonian packing. Mathematika, 20 (1973), 170–174.CrossRefGoogle Scholar
3.Boyd, D. W.. Improved bounds for the disk-packing constant. Aequationes Mat., 9 (1973), 99–106.CrossRefGoogle Scholar
4.Dudley, R. M.. Metric entropy of some classes of sets with differentiable boundaries. J. Approximation Theory, 10 (1974), 227–236.CrossRefGoogle Scholar
5.Federer, H.. Geometric Measure Theory (Springer-Verlag, New York, 1969).Google Scholar
6.Hawkes, J.. Hausdorff measure, entropy, and the independence of small sets. Proc. London Math. Soc. (3), 28 (1974), 700–724.CrossRefGoogle Scholar
7.Hawkes, J.. Random re-orderings of intervals complementary to a linear set. Quart. J. Math. Oxford (2), 35 (1984), 165–172.CrossRefGoogle Scholar
8.Kahane, J.-P.. Ensembles parfaits et processus de Lévy. Periodica Hungarica, 2 (1972), 49–59.CrossRefGoogle Scholar
9.Kaufman, R.. Entropy, dimension, and random sets. Quart. J. Math. Oxford (2), 38 (1987), 77–80.CrossRefGoogle Scholar
10.Kolmogorov, A. N. and Tihomirov, V. M.. ε-entropy and ε-capacity of sets in functional spaces. Amer. Math. Soc. Transl., 17 (1961), 277–364.Google Scholar
11.Larman, D. G.. On the exponent of convergence of a packing of spheres. Mathematika, 13 (1966), 57–59.CrossRefGoogle Scholar
12.Larman, D. G.. On the Besicovitch dimension of the residual set of arbitrarily packed disks in the plane. J. London Math. Soc., 42 (1967), 292–302.CrossRefGoogle Scholar
13.Melzak, Z. A.. Infinite packings of disks. Canadian J. Mathematics, 18 (1966), 838–852.CrossRefGoogle Scholar
14.Vitushkin, A. G.. Transmission and processing of information (Pergamon Press, London, 1961).Google Scholar