Publication Date
9-3-2013
Abstract
This thesis examines the history and some major results of the Gauss Circle Problem. The goal of the Gauss Circle Problem is to determine the best bound for the error between the number of lattice points inside a disk and that disks area, otherwise known as the lattice point discrepancy. First we state some of the required definitions and properties from Fourier analysis that will be used throughout. In particular, we establish asymptotic results for oscillatory integrals and more specifically for Bessel functions. After examining the geometrical method for precisely counting the number of lattice points inside a disk of radius R, we use the Poisson Summation Formula and the Bessel function results to prove initial bounds on the lattice point discrepancy. We present two such results, employing a similar technique for both, and then apply oscillatory integral asymptotics to extend this method and establish a lattice point discrepancy result for strongly convex domains.
Degree Name
Mathematics
Level of Degree
Masters
Department Name
Mathematics & Statistics
First Committee Member (Chair)
Matthew D. Blair
Second Committee Member
Michael Nakamaye
Third Committee Member
Maria Cristina Pereyra
Language
English
Keywords
Lattice theory, Convex domains, Fourier analysis, Bessel functions.
Document Type
Thesis
Recommended Citation
Brooks, Dusty. "Lattice points in disks and strongly convex domains." (2013). https://digitalrepository.unm.edu/math_etds/5
