Counting and Sieving in Group Orbits

数素数是数论中的一类问题，它已经被研究了几个世纪：其中最基本的体现之一是确定小于给定数的素数的近似数量，其答案是素数定理。我们可以考虑这类问题的高维版本：例如，有多少个斜边不超过z的毕达哥拉斯三元组（x^2+y^2=z^2的正整数）的面积不超过10个素数因子？这个特殊的问题可以用计数点（x,y,z）来表达，其中xy/2在作用于（3,4,5）的某个群的轨道上最多有10个素数因子。在这个问题中，所涉及的群体是“大”的，人们可以用经典的方法来接近它。然而，在底层群体“稀薄”的情况下（就像阿波罗包装的美丽理论一样），人们必须求助于更现代的工具，即布尔甘-甘伯德-萨尔纳克在2011年开发的仿射筛。PI计划不仅要研究特定有趣群（如阿波罗群）轨道的算术性质，而且要研究一般瘦群的性质：例如，如何判断一个给定的矩阵群是否瘦？一个随机有限生成的矩阵群应该是薄的吗？这些问题涉及不确定性，需要复杂的工具组合来解决，这些工具来自不同的领域——几何、数论、组合学。此外，PI还建议开发几个计算机程序，以高精度地确定其中一些问题和相关问题的答案。GL（n，Z）的瘦子群是指GL（n，C）中其Zariski闭包的Z点具有无穷指数的子群。相对于在自同构形式理论中普遍存在的与瘦群对应的算术群，在瘦群数论的应用中有许多核心问题没有得到解答。这样一个紧迫的问题是，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具有普遍性。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试图在薄单群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的固有几何，在瘦群上证明各种定理，这些定理将使它们与已知的算术群更加一致。PI还试图开发各种计算机算法，以预测给定其生成器的群体的各种属性，从扎里斯基密度到厚度。请记住，当前对瘦群感兴趣的动机源于数论，PI还建议研究瘦群的算术方面，将PI先前关于阿波罗群的一些结果推广到更大的群类。

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.