Problems in Computational Number Theory

我研究的主要重点是数论。通常我处理的问题带有计算或分析的味道。我对离散几何、遍历理论、动力系统和分形有额外的兴趣。下面我列出了一些我感兴趣的问题类型。这绝不是一个全面的清单。代数整数是具有整数系数的单多项式的根。代数整数有许多特殊的类别，它们一直是许多研究的焦点。例子包括环切数、皮索特数和塞勒姆数。关于它们的马勒测度、β展开、谱和伯努利卷积的性质，许多这类仍然没有完全理解。我研究的一个重要部分一直与这些特性有关。预计这种情况将继续下去。这些研究有各种各样的应用。这一领域的许多问题对本科生或刚开始学习的研究生来说都是可以接触到的，并且往往是HQP（高素质个人）研究的重点。第二个例子，设q是一个大于1的正整数。我们用s_q(n)表示n的第q位数字之和。例如，如果q = 10，则s_10(1729) = 1 + 7 + 2 + 9 = 19。近年来，为了更好地理解s_q在某些正整数子序列上的分布性质，人们做了很多努力。特别注意的是s_q(p(n))/s_q(n)，其中p(n)是一个整数系数的多项式。这个比率能达到什么样的值，我还没有完全理解，这是我打算进一步研究的问题。其中一个主要的公开猜想是这个比率的平均值。这一领域的许多问题对于本科生或刚开始学习的研究生来说都是可以接触到的，并且往往是HQP研究的重点。最后一个例子，考虑一个N x N个矩阵的集合。如果我们从这个集合中取矩阵的长乘积，一个自然的问题是：这个增长有多快？更具体地说，矩阵的长积的矩阵范数的上值能增长多快？第二个问题可能是：一个“大”产品是什么样的？例如，一个“大”产品是由一个特定的矩阵主导的，还是由多次出现的矩阵模式主导的？这类问题在一组矩阵的联合谱半径的研究中已经被研究过。这是一个非常活跃的研究领域，有许多悬而未决的重要问题。大部分研究将由尼基塔·西多罗夫进行。这项建议的一个重要部分是培训高素质的人才。在培养本科生、研究生和博士后水平的学生方面，我取得了很好的成功。我有很多项目的想法，只是在等待合适的学生。

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.