Dissertations, Theses, and Capstone Projects

Date of Degree

9-2023

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Alexander Gamburd

Committee Members

Enrique Pujals

Victor Kolyvagin

Subject Categories

Dynamical Systems | Harmonic Analysis and Representation | Number Theory | Other Mathematics

Abstract

In 1998 Charles Radin and John Conway introduced the Quaquaversal Tiling. A three dimensional hierarchical tiling with the property that the orientations of its tiles approach a uniform distribution faster than what is possible for hierarchical tilings in two dimensions. The distribution of orientations is controlled by the spectrum of a certain Hecke operator, which we refer to as the Quaquaversal Operator. For example, by showing that the largest eigenvalue has multiplicity equal to one, Charles Radin and John Conway showed that the orientations of this tiling approach a uniform distribution. In 2008, Bourgain and Gamburd showed that this operator has a spectral gap, which implies that the measure approaches equidistribution exponentially fast. Earlier, in 2006, Draco, Sadun and Van Wieren studied the spectrum of this operator numerically and observed that this operator has eigenvalues very close to one. Which implies that, while the orientations of the tiles converge to uniformity at an exponential rate- this rate is quite slow. In the course of their numerical experiments they observed and conjectured that the spectrum of this operator is real. This was quite striking, as the operator is not Hermitian and there seems to be no reason for it to have a real spectrum whatsoever. Here we prove this conjecture. To show this, we block triangulate the operator with respect to a well chosen partition and check that the blocks along the main diagonal are Hermitian. After we resolve the conjecture, the block partition is used to analyze the spectrum of three of the four blocks resulting in an exact description of about three quarters of the eigenvalues. We then discuss the remaining block and make a few conjectures. In the last chapter we discuss a few related results due to Conway and Radin and a result due to Serre.

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