Abstract
In this thesis, we study a variety of problems at the interface of arithmetic combinatorics, analytic number theory and harmonic analysis. We begin by considering the topic of Vinogradov's mean value theorem, an important subject of research that has seen decisive progress in the past decade via the decoupling programme pioneered by Bourgain--Demeter--Guth, and the efficient congruencing method developed by Wooley. Here, we analyse the number of solutions of the quadratic Vinogradov system of equations, when all the variables are restricted to lie in sparse sets of real numbers, a setting where both the aforementioned techniques provide ineffective estimates. In this case, we supply new bounds that improve upon and generalise work of Bourgain and Demeter. We also utilise this circle of ideas to tackle the problem of studying additive equations over lattice points on the sphere in three and four dimensions. Our results provide threshold breaking estimates which make progress towards a conjecture of Bourgain and Demeter. Both these works utilise a blend of ideas from arithmetic combinatorics, incidence geometry and analytic number theory, including the recently developed techniques known as the higher energy method.\par
We also study inverse theorems asociated with the Vinogradov system of equations. In particular, given a set $A$ of integers with extremally large number of solutions to the Vinogradov system, we prove that the set $A$ enjoys many other arithmetic combinatorial properties of interest, such as having a large arithmetically structured subset. This can be interpreted as a non--linear generalisation of a classical result in additive combinatorics, known as the Balog--Szemer\'edi--Gowers theorem. This further enables us to show that the set $A$ exhibits many distinct pairwise products, thus producing a higher degree analogue of the sum--product phenomenon.
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We also study the sum--product problem in the case of sets of diagonal matrices, where we provide new quantitative estimates, that, in this setting, improve upon previous work of Chang. This can be inferred as a natural generalisation of the sum--product phenomenon to $\mathbb{R}^d$, wherein, the product of two vectors is defined coordinate wise. Continuing this theme of problems, we also investigate lower bounds for various types of sumsets in higher dimensions. In particular, we present new results on cardinalities of difference sets and sums of dilates in $\mathbb{R}^d$, which strengthen previous results of Balog--Shakan and Stanchescu, and make progress towards a conjecture of Ruzsa. We also use these ideas to resolve the first set of non-trivial cases of a conjecture of Bukh, which concerns lower bounds for cardinalities of sumsets of the form $A +\mathscr{L}(A)$, where $\mathscr{L}$ is some linear transformation from $\mathbb{R}^d \to \mathbb{R}^d$.
| Date of Award | 28 Sep 2021 |
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| Original language | English |
| Awarding Institution |
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| Supervisor | Andrew R Booker (Supervisor), Trevor Wooley (Supervisor) & Nina C Snaith (Supervisor) |