Modern Applications of the Circle Method

Abstract

In this thesis we use recent versions of the Circle Method to prove three theorems in the area of additive number theory. It is conjectured that every even integer except two is the sum of two primes. We do not know how to prove this yet. However, we can show that the set of exceptions has zero density. Our first result is that this stays true if one considers sums of two primes that lie in some sufficiently dense subset and are well distributed in residue classes. To show this, we use Green's transference Circle Method and some additive combinatorics. If we allow an additional variable, that is consider three primes, we can say much more. Vinogradov proved that every large odd integer is the sum of three primes. The central result of the thesis is to generalize this work to a sparse set of primes, the so-called Fouvry-Iwaniec primes. These are the primes that are sum of a square and a prime square. We use the the transference Circle Method as described by Maynard, Matomäki, and Shao in their work on Vinogradov’s Theorem with almost equal summands. As in the paper of Fouvry and Iwaniec a big role is played by dissections into Type I and II sums and stepping into the Gaussian Integers. We also require different techniques in the area of sieve theory such as the Rosser-Iwaniec beta sieve and Chen's sieve switching. Finally we consider the number of representations of squares by non-singular homogeneous cubic polynomials in six or more variables. For this result we use Heath-Brown's delta Circle Method, Hooley's considerations for counting zeroes of cubic forms, and deep results in the area of algebraic geometry based on Deligne's solution of the Riemann Hypothesis for varieties over finite fields