Author

Dusty Brooks

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 Advisor

Blair, Matthew D.

First Committee Member (Chair)

Nakamaye, Michael

Second Committee Member

Pereyra, Maria Cristina

Language

English

Keywords

Lattice theory, Convex domains, Fourier analysis, Bessel functions.

Document Type

Thesis

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