Ph.D Thesis


Ph.D StudentGath Yoav Avraham
SubjectCounting Lattice Points on H-Type Groups
DepartmentDepartment of Mathematics
Supervisor PROF. Amos Nevo


Abstract

The lattice point counting problem on the Heisenberg groups seeks to obtain an upper bound (as tight as possible) for the difference between the volume of a Cygan-Kor ́anyi ball of large radius, and the number of integer lattice points which lie inside the ball. This lattice point counting problem may be viewed as a non-commutative analogue of the classical lattice point counting problem for Euclidean balls. While the Euclidean case has been continuously and extensively studied for the past decades, and has been the driving force behind much of the developments in the area of analytic number theory in the 20th century, the Heisenberg case has remained untouched until the work of Rahul Garg, Amos Nevo and Krystal Taylor. In their work, Garg, Nevo and Taylor investigated the lattice point counting problem for norm balls on the Heisenberg groups for a certain family of radial Heisenberg-homogeneous norms which includes the Cygan-Kor ́anyi norm, and proved upper bound estimates for the corresponding error terms. A main question that arises is whether the upper bounds obtained by Garg, Nevo and Taylor are sharp, and if this is not the case, then it is natural to ask how can we improve upon these upper bounds and what should be the conjectural growth rate of the error term. In order to properly address these questions, this thesis seeks to expand the research into the lattice point counting problem on the Heisenberg groups by considering additional aspects related to the behavior of the error term. In particular, and in addition to re-examining existing upper bounds for the error term, we shall also consider the problem of establishing moment and omega estimates, as well as other problems related to the fluctuating nature of the error term.