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.

在不断增长的整数集中计算素数或素数很少的数字是数论中的一类问题，已经研究了几个世纪：这个问题最基本的体现之一是大约确定小于给定数的素数的数量，并且它的答案是素数定理。 人们可以考虑此类问题的高维版本：例如，有多少个斜边最多为 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 个素数因子？ 这个特定问题可以用计数点 (x,y,z) 来表达，其中 xy/2 在某个群作用于 (3,4,5) 的轨道上最多有 10 个素因子。 这个问题涉及的群体比较“大”，可以用经典的方法来解决。 然而，在底层群很“薄”的情况下（正如美丽的阿波罗堆积理论中那样），我们必须求助于更现代的工具，即 Bourgain-Gamburd-Sarnak 于 2011 年开发的仿射筛。 PI建议不仅要研究特定有趣群（例如阿波罗群）的轨道算术性质，还要研究一般薄群的性质：例如，人们如何分辨如果给定的矩阵群很薄？ 人们是否应该期望随机有限生成的矩阵组很薄？ 这些问题具有不可判定性，需要几何、数论、组合学等不同领域的工具的复杂组合来解决。 此外，PI 建议开发几个计算机程序来高精度地确定其中一些问题及相关问题的答案。GL(n,Z) 的薄子群是那些在其 Zariski 闭包的 Z 点上具有无限索引的子群在 GL(n,C) 中。 与算术群相反，即在自守形式理论中普遍存在的薄群对应物，薄群上还有许多未解答的核心问题，这些问题对于薄群数论的应用至关重要。 一个紧迫的问题是如何判断给定的有限生成群是否是瘦群，以及瘦群在某种意义上是否是泛型的。 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 试图在薄单性组的子类中回答这些问题。 此外，PI提出的程序将深入研究所讨论的群固有的几何形状，证明薄群的各种定理，这将使它们更符合算术群的已知知识。 PI 还寻求开发各种计算机算法，在给定生成器的情况下预测群的各种属性，从 Zariski 密度到稀疏度。 请记住，当前对瘦群的兴趣源于数论，PI 还建议在瘦群的算术方面开展工作，将 PI 之前关于阿波罗群的一些结果推广到更大的群类。