Topics in sieve theory

Thesis

This thesis is concerned with applications of sieve methods to two Diophantine problems.

The first problem requires us to prove the existence of a certain infinite configuration of sums of two squares. Namely, the aim is to exhibit two increasing sequences of integers ai and mj such that mj − ai = □ + □ for every 1 ≤ i ≤ j. This set-up has a geometric interpretation which arises naturally in the field of quantum chaos. We tackle this problem using a modified half-dimensional Maynard-Tao sieve, and present details of the argument in Chapter 2.

The second problem asks us to show that a certain Diophantine equation has few integer solutions. We approach this problem by adapting the polynomial sieve developed by Browning and using the work of Weil and Deligne to obtain appropriate estimates for various exponential sums which naturally arise. As a corollary to the main result, we will deduce that polynomials with integer coefficients have “small asymmetric additive energy,” a result which has various applications in the literature. We give the details of this argument in Chapter 3.

Authors: McGrath, O

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: analytic number theory; sieve methods
• Subjects: Mathematics