Applications of the Hardy-Littlewood Method in Number Theory and Beyond

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

This proposal highlights the role of the Hardy-Littlewood (circle) method as a flexible tool that underpins progress in many areas of mathematics intersecting with number theory. The proposer intends to pursue a number of investigations that illustrate this intersubdisciplinary aspect of the method, concentrating on applications in arithmetic geometry and combinatorics. First, the researcher constructs a large class of examples of algebraic varieties, of relatively small dimension in terms of their degree, that may be studied by means of the circle method. Questions concerning the density of rational points and weak approximation should be accessible, the intention being to shed light on conjectures of Manin and others. In a second direction, the proposer applies results from the modern theory of diophantine approximation in order to refine Gower's recent explicit version of Szemeredi's theorem. In a more traditional line of enquiry, the researcher investigates the distribution of the number of representations of integers as sums of powers. The latter work enables higher moments than the traditional mean square to be considered, leading to multidimensional applications of previously inaccessible type. Number Theory studies the properties of integers (``whole numbers''). Since antiquity, the study of diophantine equations (equations to be solved in integers) has formed a core component of Number Theory, and has recently influenced the development of codes and cryptosystems (applied, for example, in data storage systems such as compact disks and DVDs, communications systems and banking security). The investigation of moment problems is a fundamental topic for Fourier analysis, which in the larger setting plays a crucial role in electrical engineering and communications. The circle method, in its modern incarnations, provides a mechanism for transferring technology from one of these areas to another, and this proposal pursues basic research that facilitates such transfers.