Additive structure in convex sets and related topics

Abstract

This thesis predominantly discusses a handful of problems in additive combinatorics and incidence geometry.
Our particular interest within additive combinatorics is proving sumset and energy bounds for convex sets and images of structured sets under convex functions. For both types of problems, we extend existing techniques and pioneer new ones for longer sums and functions with higher convexity.
The results and approaches yield applications in short sumset and energy estimates, few product – many sum problems, and counting lattice points on convex curves.
The secondary focus of this thesis is proving incidence results in the setting of thickened points and lines (atoms and tubes). We prove a result reminiscent of the Szemerédi–Trotter Theorem and applications, while also addressing the key modifications required to adapt results from traditional incidence geometry.

Date of Award	27 Sept 2022
Original language	English
Awarding Institution	University of Bristol
Supervisor	Misha Rudnev (Supervisor)