Counting and Sieving in Group Orbits

群轨道中的计数和筛分

• 批准号: 1664298
• 负责人: Elena Fuchs
• 金额: $ 6.37万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-26 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1664298&HistoricalAwards=false
• 关键词: Counting; Sieving; Group; Orbits

Counting primes or numbers with few prime factors in growing sets of integers is a class of problem in number theory which has been studied for centuries: one of the most basic incarnations of this is determining approximately the number of primes less than a given number, and its answer is the prime number theorem. One can consider higher dimensional versions of this type of question: for example, how many Pythagorean triples (positive integers with x^2+y^2=z^2) with hypotenuse at most Z have area with at most 10 prime factors? This particular question can be phrased in terms of counting points (x,y,z) for which xy/2 has at most 10 prime factors in an orbit of a certain group acting on (3,4,5). In this problem, the group involved is "big" and one can use classical methods to approach it. However, in the case where the underlying group is "thin" (as it is in the beautiful theory of Apollonian packings), one must appeal to much more modern tools, namely the Affine Sieve developed by Bourgain-Gamburd-Sarnak in 2011. The PI proposes to study not only the arithmetic properties of orbits of specific interesting groups (such as the Apollonian group), but also to investigate properties of thin groups in general: for example, how does one tell if a given matrix group is thin? Should one expect a random finitely generated matrix group to be thin? These questions toy with undecidability and require an intricate combination of tools from various fields -- geometry, number theory, combinatorics -- to tackle. In addition the PI proposes to develop several computer programs to determine the answers to some of these and related questions with high accuracy.Thin subgroups of GL(n,Z) are those which are of infinite index in the Z-points of their Zariski closure in GL(n,C). In contrast to arithmetic groups, the counterparts to thin groups which are prevalent, say, in the theory of automorphic forms, there are many unanswered core questions on thin groups which are essential in applications to the number theory of thin groups. A pressing such question is how to tell if a given finitely generated group is thin, as well as whether thin groups are generic in some sense. These two questions have been answered in a few situations, and the PI proposes to answer them in much higher generality. The PI seeks to answer these questions in the subclass of thin monodromy groups. Furthermore, the PI's proposed program will delve deeply into the geometry inherent to the groups in question, proving various theorems on thin groups which will bring them more in line with what is known on arithmetic groups. The PI also seeks to develop various computer algorithms which would predict various properties of a group given its generators, from Zariski density to thinness. Keeping in mind that the motivation for the current interest in thin groups stems from number theory, the PI also proposes to work on the arithmetic side of thin groups, generalizing some of the PI's previous results about the Apollonian group to a much larger class of groups.

在增长的整数集合中计算素数或具有少数素数因子的数是数论中的一类问题，已经研究了几个世纪：这一问题的最基本体现之一是近似确定小于给定数的素数的数量，其答案是素数定理。 我们可以考虑这类问题的更高维版本：例如，有多少个最多使用Z的毕达哥拉斯三元组（x^2+y^2=z^2的正整数）的面积最多有10个素因子？ 这个特殊的问题可以用计数点（x，y，z）来表述，对于这些点，xy/2在某个群作用于（3，4，5）的轨道上最多有10个素因子。 在这个问题中，所涉及的群是“大”的，人们可以使用经典的方法来处理它。然而，在基础群是“薄”的情况下（就像在美丽的阿波罗填充理论中一样），人们必须求助于更现代的工具，即由Bourgain-Gamburd-Sarnak在2011年开发的仿射筛。 PI建议不仅要研究特定有趣群（如阿波罗群）轨道的算术性质，还要研究一般薄群的性质：例如，如何判断一个给定的矩阵群是否薄？ 人们是否应该期望一个随机生成的矩阵群是稀疏的？ 这些问题具有不可判定性，需要几何学、数论、组合学等各个领域的工具进行复杂的组合才能解决。 此外，PI建议开发几个计算机程序，以确定答案的一些这些和相关问题的高精度。薄子群GL（n，Z）是那些是无限指数的Z-点，其Zapriki封闭GL（n，C）。 与算术群相反，瘦群的对应物在自守形式理论中很普遍，瘦群有许多未回答的核心问题，这些问题在瘦群的数论应用中是必不可少的。 一个紧迫的问题是如何判断一个给定的随机生成的组是否是瘦的，以及瘦组在某种意义上是否是通用的。 这两个问题已经在一些情况下得到了回答，PI建议以更高的普遍性来回答它们。 PI试图在薄单值群的子类中回答这些问题。 此外，PI提出的计划将深入研究所讨论的群所固有的几何，证明瘦群上的各种定理，这将使它们与算术群上的已知定理更加一致。 PI还寻求开发各种计算机算法，这些算法可以预测给定生成元的群的各种属性，从Zebraki密度到薄度。 考虑到当前对瘦群的兴趣源于数论，PI还建议研究瘦群的算术方面，将PI以前关于阿波罗群的一些结果推广到更大的一类群。