The purpose of this proposal is to illustrate and exploit the observation that the Hardy-Littlewood (circle) method is a fundamental technique of arithmetic harmonic analysis that provides a powerful interface connecting arithmetic geometry, additive combinatorics and harmonic analysis. First, in arithmetic geometry, the proposer applies diagonalization techniques to investigate the density of rational points on complete intersections of high dimension, applies the large sieve inequality to study the Hasse principle in algebraic families of complete intersections, and combines the circle method with descent techniques so as to investigate rational points on varieties of intermediate dimension. On the interface with harmonic analysis, the proposer investigates the Fourier transform of the solution set of polynomial diophantine inequalities in many variables, with intended applications to uniform distribution and polynomial ergodic results. Third, on the interface with additive combinatorics, the proposer seeks to establish a novel version of the circle method that exploits Gower's quadratic and higher uniformity ideas. In one direction,applications here would impact questions in arithmetic geometry, and in another the proposer seeks quantitative improvements in the work of Gowers and Green-Tao by applying techniques from the modern theory of the circle method.
This proposal investigates the interface between three areas of mathematics: number theory (and more specifically diophantine problems), harmonic analysis and additive combinatorics. 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 internet and web-based commerce). Harmonic analysis investigates generalizations of Fourier analysis, which in the larger setting plays a crucial role in electrical engineering and communications. Additive combinatorics seeks to understand the underlying structure in quite general (and seemingly, therefore, unstructured) sets, particularly as these sets are modified by arithmetic operations. This proposal applies a fundamental technique known as the circle method to transfer technology between these three areas, both enhancing our knowledge in each area and increasing the scope of the circle method as a basic tool of arithmetic harmonic analysis.
