Solving nonlinear equations in arithmetic sets.

This is a project in additive number theory. This subfield of number theory broadly concerns the study of subsets of integers and their behaviour under addition. It has close ties to prime number theory, combinatorial number theory and the geometry of numbers. Two classical problems in the field are the Goldbach conjecture (which is the conjecture that every even number greater than two is a sum of two primes) and Waring's problem (which asks whether, for a given integer k, every integer can be expressed as a sum of a bounded number of k-th powers). Many of these problems are studied using the tools from the Hardy-Littlewood circle method and from sieve methods. Often the set of prime numbers is of particular interest and these techniques have been applied successfully to several related problems. For example, Vinogradov used them to prove that every sufficiently large odd number is the sum of three primes. In a similar spirit, this project will aim to count solutions to certain higher-degree equations in the prime numbers, but due to the non-linear behaviour one has to go beyond the classical methods mentioned above. Hence, the project has interfaces with other areas such as representation theory and combinatorics. It is a project in pure mathematics and no impact outside of the subject is currently envisaged.

The initial aim of the project is to look more closely at several aspects of a very recent paper by my supervisor Ben Green. In this paper, methods of group theory and representation theory are used to determine the asymptotic number of solutions to generic quadratic equations in 8 variables, where the variables are required to be prime. Various natural questions present themselves. For instance, can these methods be adapted to sets of integers other than the primes, perhaps even to essentially arbitrary sufficiently dense sets? Can the methods be extended to handle more general equations than homogeneous quadratics, for example by allowing linear terms? Does this approach require the current genericity assumption on the quadratic form, or can it be removed with further work? Subsequent aims of the project would be to look in greater generality at the potential for applying the representation theory of finite groups to questions in additive number theory, or to look at generic equations of degree greater than 2.

The paper of Green is very recent (August 2021) and the methodology in that paper is quite novel. Indeed, the paper appears to be the first application of the representation theory of finite groups to the circle method. Hence, these methods extend traditional tools for studying additive questions in number theory, and it is worth investigating their general applicability to additive number theory problems.

This project falls within the EPSRC Number Theory research area.