Counting and Sieving in Group Orbits

群轨道中的计数和筛分

• 批准号: 1501970
• 负责人: Elena Fuchs
• 金额: $ 15.2万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-15 至 2016-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1501970&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.

在成长中的整数中计数数字或数量很少的数字是数字理论的一类问题，它已经进行了数百年的研究：其中最基本的化身之一是确定大约小于给定数字的数量的数量，并且它的答案是质数定理。 可以考虑此类问题的更高维版本：例如，最多Z的pythagorean三元组（带有X^2+Y^2 = Z^2的正整数）在最多有10个素数因素？ 这个特定的问题可以用计数点（x，y，z）来表达，xy/2在某个轨道的轨道中最多有10个主要因素（3,4,5）。 在此问题中，所涉及的小组是“大”，并且可以使用经典方法来处理它。 但是，在基础群体“薄”的情况下（就像Apollonian包装的美丽理论一样），必须吸引更多现代工具，即2011年Bourgain-Gamburd-Sarnak开发的仿射筛。 PI建议不仅研究特定有趣组的轨道（例如阿波罗尼亚组）的算术特性，而且还研究了一般的薄基团的特性：例如，如何确定给定基质组是否薄？ 应该期望有限生成的矩阵组稀薄吗？ 这些问题的玩具具有不可证明的性能，需要从各个领域的工具（几何学，数理论，组合学）来解决的复杂组合来应对。 此外，PI建议开发多个计算机程序，以高精度确定其中一些和相关问题的答案。GL（N，Z）的thin子组是Zariski封闭Z点中无限索引的子组。在GL（N，C）中。 与算术群体相反，与薄群相比，在薄类的理论中，薄类普遍存在，薄群的核心问题有许多未解决的核心问题，这对于对薄群的数量理论的应用至关重要。 一个迫切的问题是如何判断给定有限生成的组是否薄，以及薄群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建议以更高的一般性回答它们。 PI试图在薄单构小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组固有的几何形状，证明了薄群的各种定理，这将使它们更符合算术组所知。 PI还试图开发各种计算机算法，这些计算机算法将预测一组发生器的各种特性，从Zariski密度到薄度。 请记住，当前对薄群体兴趣的动机源于数字理论，PI还建议在薄群的算术方面工作，从而将PI先前关于Apollonian群体的一些结果推广到更大的群体中。