Problems in Computational Number Theory

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The primary focus of my research is in number theory. Typically I deal with problems with a computational or analytic flavour. I have additional interests in discrete geometry, ergodic theory, dynamical systems and fractals. Below I have listed some of the types of problems that I am interested in. This is in no way a comprehensive list.**An algebraic integer is the root of a monic polynomial with integer coefficients. There are many special classes of algebraic integers that have been the focus of much research. Examples include cyclotomic numbers, Pisot numbers and Salem numbers. Properties concerning their Mahler measure, beta-expansions, spectra and Bernoulli convolutions are still not fully understood for many of these classes. A significant part of my research has historically been concerned with these properties. This is expected to continue. These investigations have a wide variety of applications. Many problems in this area are accessible to undergraduate or starting graduate students, and will often be the focus of HQP (Highly Qualified Personal) research.**As a second example let q be a positive integer greater than 1. We denote by s_q(n) the sum of the q-ary digits of n. For example, if q = 10, then s_10(1729) = 1 + 7 + 2 + 9 = 19. In recent years, much effort has been made to get a better understanding of the distribution properties of s_q regarding certain subsequences of the positive integers. Particular attention has been given to the ratio s_q(p(n))/s_q(n) where p(n) is a polynomial with integer coefficents. Questions concerning which values this ratio can achieve are still not fully understood, and is something I plan to futher investigate. One of the main open conjectures about this is the average value of this ratio. Many problems in this area are accessible to undergraduate or starting graduate students, and will often be the focus of HQP research.**As a last example, consider a set of N x N matrices. If we take long products of matrices from this set, one natural question to ask is: how quickly does this grow? More specifically, how quickly can the supremum of the matrix norm of long products of matrices grow? A second question might be: What does a "big" product look like? For example, is a "big" product dominated by a particular matrix, or pattern of matrices occurring multiple times? These sorts of questions have been looked at in the study of the joint spectral radius of a set of matrices. This is a very active area of research, with many unanswered and important questions. Much of this research will be conducted with Nikita Sidorov.**A significant part of this proposal is for the training of highly qualified personal. I have had good success with training students, at the undergrad, graduate and postdoctoral level. I have numerous ideas for projects that are just waiting for the right student.