UBC Theses and Dissertations

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UBC Theses and Dissertations

Open disk packings of a disk Wilker, John Brian

Abstract

A packing of the plane unit disk U by an infinite collection of smaller disks [symbol omitted] = {Dn} is a non-over lapping arrangement of the Dn which covers U up to a residual set of measure 0. An indication of the efficiency of such a packing is given by its exponent and local exponents which are defined in terms of the convergence of the exponential series [formula omitted] where rn is the radius of Dn and α is positive. It is proved that the exponent of a packing is the supremum of its local exponents. Then a special class of packings is introduced and it is shown that all these have the same exponent and constant local exponent. Reasons are given for believing this exponent to be the minimum over all packings and a lower bound of 1.059 is derived for it. One of these packings is modified without changing its exponent to solve an obstacle problem. In the final section, several unsolved problems on packings and exponents are suggested.

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