Open Collections

White, Ethan Patrick

University of British Columbia

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.

Text

2023-04-21

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/84416

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/