Additive Combinatorics

Over the course of my DPhil, I intend to focus on combinatorial problems with an additive or number theoretic flavour. Initially, a problem I plan to spend some time thinking about is motivated by a recent result by Fox and Lovász (refining a theorem by Green) stating that subsets of large finite vector spaces with few additive triples x + y = z can be reduced in size by a small amount to remove all additive triples. Green's original result gave weak bounds for the amount the subset must be reduced, and Fox and Lovász improved these bounds dramatically. Green's original paper applied to arbitrary abelian groups, whereas Fox and Lovasz' work required the group to be a vector space. The problem, then, is to see if any progress can be made applying these ideas to arbitrary abelian groups. In particular, the method Fox and Lovász used involved a result of Kleinberg, which gave an upper bound for the size of sets X, Y and Z with the property that each element of one of those sets is in exactly one additive triple of the form x + y = z with each term taken from the obvious set. Kleinberg's result came with polynomial bounds, leading to polynomial bounds in the removal lemma. Such a result is provably too strong for arbitrary abelian groups, but a potential approach would be to see how a modification of Kleinberg's result could impact the result on removing additive triples in general groups.

A second problem I intend to think about relates to so-called Generalised Von Neumann theorems. These theorems are results that show that if the average product of some functions applied to numbers which satisfy a linear relation is large, then that must be because one of the functions has a large Gowers norm. The obvious next step is to consider how this is affected when, instead of looking at numbers satisfying a linear relation, we consider numbers satisfying a quadratic form, and, more specifically, what the correct analogue of the Gowers norm should be in this case. A particularly pertinent example of such functions would be to take the indicator function of the set of primes; in that case, bounds for this problem would transfer to statements about when there are solutions to the quadratic form in which all of the coordinates are prime.

These results all fall under the general umbrella of additive combinatorics, which has strong links to analytic number theory. In particular, they can tell us facts about prime numbers, which have great importance to fields such as cryptography and computer science, as well as being interesting in their own right.


This project falls within the EPSRC Number Theory research area.