The residual set dimension of the Apollonian packing

In this paper we show that, for the Apollonian or osculatory packing C 0 of a curvilinear triangle T , the dimension d ( C 0 , T ) of the residual set is equal to the exponent of the packing e ( C o , 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 ( C 0 , T ). In practical terms it is thus now known that 1·300197 d ( C 0 , T )

[1]  H. Eggleston.  On closest packing by equilateral triangles , 1953, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  H. Furstenberg,et al.  Products of Random Matrices , 1960 .

[3]  D. Larman,et al.  On the exponent of convergence of a packing of spheres , 1966 .

[4]  Z. A. Melzak.  Infinite Packings of Disks , 1966, Canadian Journal of Mathematics.

[5]  K. Hirst.  The Apollonian Packing of Circles , 1967 .

[6]  D. Larman.  On the Besicovitch Dimension of the Residual Set of Arbitrarily Packed Disks in the Plane , 1967 .

[7]  J. Wilker.  Open Disk Packings of a Disk , 1967, Canadian Mathematical Bulletin.

[8]  N. Tschoegl,et al.  Osculatory Packing of Finite Areas with Circles , 1970, Nature.

[9]  D. Boyd.  Osculatory Packings by Spheres , 1970, Canadian Mathematical Bulletin.

[10]  D. Boyd.  Lower Bounds for the Disk Packing Constant , 1970 .

[11]  David W. Boyd,et al.  The disk-packing constant , 1971 .

[12]  D. Boyd.  Disk Packings which have Non-Extreme Exponents , 1972, Canadian mathematical bulletin.

[13]  D. Boyd.  Improved bounds for the disk-packing constant , 1973 .