Topics in analytic and combinatorial number theory by Walker, A

“…<p>In this thesis we consider three different issues of analytic number theory. Firstly, we investigate how residues modulo <em>q</em> may be expressed as products of small primes. …”

Topics in combinatorics and additive number theory by Aaronson, JA

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Three topics in additive prime number theory by Green, B

Some results at the interface of combinatorics and number theory by Chase, Z

Subjects:

Quadratic phenomena in additive combinatorics and number theory by Lindqvist, S

“…</p> <p>In Chapter 2 we recall some standard results in number theory and additive combinatorics.</p> <p>In Chapter 3 we look at partition regularity of equations of the form x^a + y^b = z^c over Z/pZ. …”

Galois representations attached to algebraic automorphic representations by Green, B

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Generalising the Hardy-Littlewood Method for Primes by Green, B

“…The Hardy-Littlewood method is a well-known technique in analytic number theory. Among its spectacular applications are Vinogradov's 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the number of 3-term progressions of primes, all less than N. …”

Approximate groups and their applications: work of Bourgain, Gamburd, Helfgott and Sarnak by Green, B

“…This is a survey of several exciting recent results in which techniques originating in the area known as additive combinatorics have been applied to give results in other areas, such as group theory, number theory and theoretical computer science. We begin with a discussion of the notion of an approximate group and also that of an approximate field, describing key results of Freiman-Ruzsa, Bourgain-Katz-Tao, Helfgott and others in which the structure of such objects is elucidated. …”

A Szemeredi-type regularity lemma in abelian groups, with applications by Green, B

“…In this paper we prove an analogue of Szemeredi's regularity lemma in the context of abelian groups and use it to derive some results in additive number theory. One is a structure theorm for sets which are almost sum-free. …”

Counting sets with small sumset, and the clique number of random Cayley graphs by Green, B

“…Despite the graph-theoretical title, this is a paper in number theory. Our main results are essentially estimates for the number of sets A in {1,...,N} with |A| = k and |A + A| = m, for various values of k and m.…”