Incidence Geometry in the Plane and Applications to Arithmetic Combinatorics

Abstract

In this thesis we study the applications of incidence geometry to additive combinatorics, with a particular focus of the setting of the prime residue field.

We begin by introducing some context to incidence geometry and its role in combinatorics. In Chapters 4 and 5, we prove a new incidence bound between points and lines in the plane and discuss its applications. The sum-product phenomenon is a motivating problem of this thesis and we discuss this in both the case of an arbitrary field and the specific case of F= R. In this latter setting, in Chapter 6 we prove a bound on the energy formulation of the sum-product problem. A consequence of this is to the sum-product problem itself, as well as to that of expander functions. Finally, in Chapter 8, we prove a new bound on the number of pinned distances in the plane.

Date of Award	24 Mar 2020
Original language	English
Awarding Institution	University of Bristol
Supervisor	Joseph Najnudel (Supervisor) & Misha Rudnev (Supervisor)