Topics in discrete analysis
department or school: Science, Faculty of
department or school: Mathematics, Department of
data provider: DSpace
degree campus: UBCV
creator: White, Ethan Patrick
issued: 2023-04-21T17:50:41Z
issued: 2023
related degree: Doctor of Philosophy - PhD
degree grantor: University of British Columbia
description: This dissertation is comprised of four articles, each related to a discrete extremal problem. Several topics appear in more than one chapter. These include polynomial methods, Fourier analysis, and combinatorial number theory. In Chapter 2 we prove that the number of directions contained in a set of the form A x B ⊂ AG(2,p) where p is prime, is at least |A||B| - min{|A|,|B|} + 2. Here A and B are subsets of GF(p) each with at least two elements and |A||B|<p. This bound is tight for an infinite class of examples. Our main tool is the use of the Rédei polynomial with Szőnyi's extension. In Chapter 3 we determine the optimal constant in a convolution inequality up to a 10⁻⁶ error. Furthermore, we prove that a unique minimizer exists. As a corollary, we obtain improvements on the maximum size of Bh(g) sets for (g,h) ∈ {(2,2),(3,2),(4,2),(1,3),(1,4)}. A combination of Fourier analysis and convex optimization is used to obtain the main result. In Chapter 4 we obtain a substantially improved lower bound for the minimum overlap problem asked by Erdős, very close to the best upper bound. Our approach uses elementary Fourier analysis to translate the problem to a convex optimization program. In Chapter 5 we study arrangements of intervals in R² for which many pairs are concyclic. We show that any set of intervals with many concyclic pairs must have underlying algebraic and geometric structure. In the most general case, we prove that the endpoints of many intervals belong to a single bicircular quartic curve.
aggregated CHO: https://circle.library.ubc.ca/rest/handle/2429/84416
note: Topics in Discrete AnalysisbyEthan Patrick WhiteM.Sc., The University of British Columbia, 2019A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)May 2023© Ethan Patrick White, 2023The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the dissertation entitled:Topics in Discrete Analysissubmitted by Ethan Patrick White in partial fulfillment of the requirements for the degreeof Doctor of Philosophy in Mathematics.Examining Committee:Jo´zsef Solymosi, Professor, Mathematics, UBCSupervisorRichard Anstee, Professor, Mathematics, UBCSupervisory Committee MemberAndrew Rechnitzer, Professor, Mathematics, UBCUniversity ExaminerBruce Shepherd, Professor, Computer Science, UBCUniversity ExaminerGa´bor Tardos, Professor, Alfre´d Re´nyi Institute of MathematicsExternal ExaminerAdditional Supervisory Committee Members:Joshua Zahl, Associate Professor, Mathematics, UBCSupervisory Committee MemberiiAbstractThis dissertation is comprised of four articles, each related to a discrete extremal problem.Several topics appear in more than one chapter. These include polynomial methods, Fourieranalysis, and combinatorial number theory.In Chapter 2 we prove that the number of directions contained in a set of the formA×B ⊂ AG(2, p), where p is prime, is at least |A||B|−min{|A|, |B|}+2. Here A and B aresubsets of GF (p) each with at least two elements and |A||B| < p. This bound is tight for aninfinite class of examples. Our main tool is the use of the Re´dei polynomial with Szo˝nyi’sextension.In Chapter 3 we determine the optimal constant in a convolution inequality up to a 10−6error. Furthermore, we prove that a unique minimizer exists. As a corollary, we obtain im-provements on the maximum size of Bh[g] sets for (g, h) ∈ {(2, 2), (3, 2), (4, 2), (1, 3), (1,…
has type: Thesis/Dissertation
date issued: 2023-05
is shown at: 10.14288/1.0431376
language: eng
degree discipline: Mathematics
provider: Vancouver : University of British Columbia Library
publisher: University of British Columbia
rights: Attribution-NonCommercial-NoDerivatives 4.0 International
rights URI: http://creativecommons.org/licenses/by-nc-nd/4.0/
scholar level: Graduate
contributor: Solymosi, Jozsef, 1959-
type: Text
identifier URI: http://hdl.handle.net/2429/84416