Award Date

5-1-2019

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

First Committee Member

Arthur Baragar

Second Committee Member

David Costa

Third Committee Member

Zhijian Wu

Fourth Committee Member

William Ramsey

Number of Pages

263

Abstract

In this thesis, we show that circle, sphere, and higher dimensional sphere packings may

be realized as subsets of the boundary of hyperbolic space, subject to certain symmetry

conditions based on a discrete group of motions of the hyperbolic space. This leads to

developing and applying counting methods which admit rigorous upper and lower bounds on

the Hausdorff (or Besikovitch) dimension of the residual set of several generalized Apollonian

circle packings. We find that this dimension (which also coincides with the critical exponent

of a zeta-type function) of each packing is strictly greater than that of the Apollonian

packing, supporting the unsolved conjecture that, among the many possible disk tilings of

the plane, the Apollonian packing has the smallest possible residual set dimension. The

obtained rigorous bounds are also consistent with the heuristic estimates calculated herein.

Keywords

Apollonius; Hausdorff dimension; Lorentz space; Sphere packing; Thin groups

Disciplines

Mathematics

Language

English


Included in

Mathematics Commons

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