Algebraic Number Fields in Fourier Analysis and Fractal Geometry

About This Grant

Fourier analysis is the study of the decomposition of functions and measures into pieces with different frequency. For smooth functions with rapid decay, this decomposition is easy to describe, but for more rough or singular objects, the decomposition is less well-behaved. This project concerns examples of measures with additional algebraic structure. Of particular interest is the case when the measure is concentrated on a small set, such as a surface or a fractal. This project will provide new insights into the relationship between problems in algebraic number theory, problems in fractal geometry, and problems in harmonic analysis. This project will provide research training opportunities for graduate students interested in fractal geometry and harmonic analysis. A common theme in Fourier analysis, analytic number theory, and geometric measure theory is the connection between estimates for oscillatory integrals, estimates for the number of solutions to a Diophantine equation, and the presence of configurations in fractal sets. This connection has been of utmost importance in the last decade, with the proof of the Vinogradov mean value theorem of Bourgain, Demeter, and Guth in three or more dimensions using decoupling estimates for the moment curve. In this project the PI seeks to further explore the connections between number theory, Fourier analysis, and fractal geometry by considering a number of new avenues: applying the theory of algebraic number fields to obtain new decoupling estimates, obtaining decoupling estimates for random Cantor sets in the real numbers, showing the optimality of the Mockenhaupt-Mitsis-Bak-Seeger restriction estimate for Orlicz spaces for higher-dimensional analogues of the well-approximable numbers, obtaining Fourier dimension estimates for number field analogues of the badly approximable numbers, obtaining prescribed projection theorems for nonlinear projections in high dimensions, and finding large subsets of Euclidean space avoiding patterns using the landmark pair framework. 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.