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Date Available

5-4-2026

Year of Publication

2026

Document Type

Doctoral Dissertation

Degree Name

Doctor of Philosophy (PhD)

College

Arts and Sciences

Department/School/Program

Mathematics

Faculty

Xuancheng Shao

Faculty

Bert Guillou

Abstract

The first part of this thesis is concerned with Goldbach-type problems. In recent years, there has been an interest in developing density versions of Goldbach-type results. Namely, given a relatively dense subset A of the primes, one may study representations of integers as sums of primes belonging to the subset A. These density Goldbach-type results have been facilitated by the development of new tools from additive combinatorics, in particular the Fourier-analytic transference principle due to Green. We apply the transference principle to obtain a variant of Vinogradov’s theorem involving subsets of primes confined to the residue class 1 (mod 3). We also develop a new variant of the transference principle and we determine conditions on a subset A of the primes that allows for almost every even integer to be represented as a sum of two primes from A. The second part of this thesis concerns the interaction between the additive and multiplicative structures of a prime field. Generalized arithmetic progressions (GAPs) are the quintessential example of an additively structured set, and there is much interest in quantifying the lack of multiplicative structure in GAPs. We are able to obtain the optimal bound on the multiplicative energy for proper GAPs of rank 2, and as an application we obtain Burgess-quality bounds for sums of Dirichlet characters over such sets. We also use character sum estimates over GAPs along with other additive combinatorial tools to study the distribution of primitive roots in Hilbert cubes, which are a class of additively structured sets.

Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2026.266

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