The Hardy-Littlewood method: explorations on the frontier

The use of Fourier series is ubiquitous throughout much of science and engineering, allowing approximately periodic phenomena to be understood and analyzed efficiently for critical applications. The Hardy-Littlewood (circle) method applies Fourier series to count solutions of Diophantine equations. Studied since ancient times, these are equations to be solved in integers, whose properties remain a central focus of modern research. In general, the circle method uses subtle aspects of Fourier series in the guise of exponential sums, tools that contribute to tests for equidistribution (apparent "randomness") of number theoretic sequences used in computer science and cryptography. Despite the recent renaissance in the Hardy-Littlewood method, the current understanding of its nuances remains plagued by basic mysteries that stand in the way of ambitious applications in analytic number theory that would be transformative in nature. The Principal Investigator will explore these frontiers of the subject, delivering progress on model problems that should enhance our understanding of the most difficult regions of the Fourier space in applications. Informed by these advances, novel approaches, some only recently established and others speculative in nature, will be employed to address fundamental problems concerning Diophantine equations, congruences and equidistribution, opening new avenues for future research. This project will also involve training graduate students in this promising new technology, and work will continue on a text making this research available to the wider mathematical community.
 
Two great mysteries of the Hardy-Littlewood method concern the nature of contributions in circle method integrals of points with large height, and the relation of special subvarieties of solutions to the associated Fourier analysis. In mean values of exponential sums over polynomials of larger degree, the provenance of points on low degree subvarieties contained in the associated hypersurfaces remains speculative. Three goals will be addressed in this project that address these mysteries. First, height-scaling techniques will be applied in the circle and related methods to obtain new conclusions concerning the representation of arithmetic sequences by higher degree forms. This will provide novel approaches to understanding the representation of integers in such sequences as the squarefree numbers by polynomials of large degree. Second, novel generalizations of the delta-function technique will be investigated, offering access to Diophantine problems of higher degree. This method offers a potential tool for the investigation of the contribution of points of medium circle method height in applications of the Hardy-Littlewood method. Subject to availability of time and resources, a third goal will be to apply congruences over p-adic function fields to derive mean value estimates for exponential sums applicable to conventional Diophantine problems.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

Award Number: 2502625
Principal Investigator: Trevor Wooley
Funds Obligated: $240,000
State: IN