Geometric and arithmetic structure in finite point sets

Abstract

We study a handful of problems in the related fields of additive combinatorics and discrete geometry. Following the prerequisite background presented in Chapters 2 and 3, the following results are given.

In Chapter 4, we prove that if a finite set of points in the real plane determines, up to multiplicative constants, the smallest possible number of congruency classes of triangles, then it must have a large intersection with a line or a circle.

In Chapter 5, we develop upon the so-called squeezing method to prove new results about iterated sum sets of images of convex functions. In particular, we improve the lower bound on the growth of the set $f(A-A)$ for a convex function $f$ belonging to a specific general class of functions.

In Chapter 6, we prove a finite field analogue of the quadratic Vinogradov mean value theorem by employing incidence geometric techniques, including a new incidence bound for a particular family of Möbius hyperbolae.

Date of Award	13 May 2025
Original language	English
Awarding Institution	University of Bristol
Supervisor	Misha Rudnev (Supervisor)