Maysum Panju's research statement for J.W.H. Zou Award

There were two main projects that I worked on during my term as an undergraduate research assistant in 2011. The original project was to classify beta expansions for regular Piscot numbers. This research was completed ahead of schedule, so I subsequently began the second project concerning the enumeration of Garsia numbers. Both projects related to computational number theory and computational algebra, and were supervised by Dr. Kevin Hare of the Department of Pure Mathematics at the University of Waterloo.

The first project concerned beta expansions, which are the analogue of the conventional base 10 expansions of real numbers where the base is allowed to be a non-whole number. By allowing non-integer bases, many familiar properties of expansions (such as termination, uniqueness or near-uniqueness, and ordering) become distorted or vanish entirely. The focus of this project was to examine how beta expansion of the real number 1 would look when the base of the expansion is taken to be a Pisot number, which is a real algebraic integer larger than 1 but with all of its Galois conjugates having modulus strictly less than 1. These peculiarly defined numbers have various interesting properties in dynamics and chaos theory, and are also known to produce relatively tame beta expansions. For this project, I completed a comprehensive classification of the patterns in these beta expansions for all Pisot numbers in the interval (1, 2), and in doing so, answered a decade-old conjecture concerning the algebraic patterns associated with these expansions.

The second project was to delve into the understanding of Garsia numbers, a previously unstudied class of numbers first mentioned by Garsia in 1962. These numbers are defined to be algebraic numbers with minimal polynomial having a norm of 2, and with all roots of the minimal polynomial having modulus strictly greater than 1. Prior to our research, the set of numbers meeting these criteria was completely mysterious: it was unknown if this set was dense or discrete, where its limit points were located, or how they could be detected and identified. A major portion of this research project was to design and implement a computer algorithm that would generate a list of all Garsia numbers up to a given fixed degree, and also generate a list of all Garsia numbers that fell within a particular closed interval (identifying any limit point that happened to lie in that region as well). The implementation of this algorithm shed some light on the distribution of Garsia numbers, including a comprehensive listing of all Garsia numbers up to degree 40, as well as general classification of an infinite family of limit points. Results from this research have already been used by other number theorists studying the interlaced roots of cyclotomic polynomials. Furthermore, techniques used in the polynomial generating algorithm can be extended to perform searches within intervals for algebraic integers satisfying similar root-based properties.