Additive Combinatorics and Diophantine Problems

Abstract

This thesis presents new results on the topics of partition regularity and density regularity. The first chapter provides an introduction to these subjects and an overview of the main results of this thesis. In Chapter 2, we study the connections between partition regularity and multiplicatively syndetic sets. In particular, we prove that a dilation invariant system of polynomial equations is partition regular if and only if it has a solution inside every multiplicatively syndetic set. We also adapt methods of Green-Tao and Chow-Lindqvist-Prendiville to develop a syndetic version of Roth's density increment strategy. This argument is then used to obtain bounds on the Rado-Ramsey numbers of configurations of the form $\{x, d, x + d, x + 2d \}$. In Chapter 3, we establish new partition and density regularity results for systems of diagonal equations in kth powers. Our main result shows that if the coefficient matrix of such a system is sufficiently non-singular, then the system is partition regular if and only if it satisfies Rado's columns condition. Furthermore, if the system also admits constant solutions, then we prove that the system has non-trivial solutions over every set of integers of positive upper density. In Chapter 4, we obtain a double exponential bound in Brauer's generalisation of van der Waerden's theorem on arithmetic progressions with the same colour as their common difference. Using Gowers' local inverse theorem, we obtain a bound which is quintuple exponential in the length of the progression. We refine this bound in the colour aspect for three-term progressions, and combine our arguments with an insight of Lefmann to obtain analogous bounds for the Rado-Ramsey numbers of certain non-linear quadratic equations. The content of this chapter is joint work with Sean Prendiville. Finally, in Chapter 5, we conclude with a summary of this thesis and describe possible directions for future research.

Date of Award	31 Dec 2021
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Donald Robertson (Supervisor) & Marcus Tressl (Supervisor)