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Full-Text Articles in Number Theory
On The Spectrum Of Quaquaversal Operators, Josiah Sugarman
On The Spectrum Of Quaquaversal Operators, Josiah Sugarman
Dissertations, Theses, and Capstone Projects
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 …
Asymptotic Counting Formulas For Markoff-Hurwitz Tuples, Ryan Ronan
Asymptotic Counting Formulas For Markoff-Hurwitz Tuples, Ryan Ronan
Dissertations, Theses, and Capstone Projects
The Markoff equation is a Diophantine equation in 3 variables first studied in Markoff's celebrated work on indefinite binary quadratic forms. We study the growth of solutions to an n variable generalization of the Markoff equation, which we refer to as the Markoff-Hurwitz equation. We prove explicit asymptotic formulas counting solutions to this generalized equation with and without a congruence restriction. After normalizing and linearizing the equation, we show that all but finitely many solutions appear in the orbit of a certain semigroup of maps acting on finitely many root solutions. We then pass to an accelerated subsemigroup of maps …