Counting and Sieving in Group Orbits

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

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.