Distance Questions, Fourier Restriction, and Beyond

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Distance questions are the driving forces in incidence geometry, a field of mathematics studying intersection patterns of basic geometric objects (such as points, lines, or circles). One of the most famous distance questions is called the Erdös distinct distance problem, which asks for the least number of distinct distances generated by a given set of points. Another example is the unit distance problem, concerning the maximum number of times that a fixed distance can occur among a given set of points. These questions have motivated development of tools and ideas that have wide applications in disciplines beyond mathematics, such as computer sciences, physics, and engineering. Fourier restriction concerns the Fourier transform, which decomposes a function into pieces with different frequencies of oscillation. Fourier restriction studies a fundamental question about Fourier transform: the relation between the size of a function and the geometry of its Fourier transform. This relation has important applications in partial differential equations, number theory, and other areas. This research project aims to further the understanding of the interplay between distance questions and Fourier restriction, as well as to develop modern tools with applications in various areas of mathematics, including harmonic analysis, geometric measure theory, and partial differential equations.One of the main directions in the project is driven by Falconer's conjecture, a continuous analogue of the distinct distance problem. It is conjectured that the distance set of a compact set E must have positive measure if the Hausdorff dimension of E exceeds a certain threshold. Recent results towards resolving the conjecture were obtained via new ideas from Fourier restriction theory: incidence geometry and geometric measure theory. This project will continue investigation of these approaches and apply them to other related distance questions such as general geometric configurations, multiparameter distances, and projections of fractal measures. The work aims to further the study of weighted Fourier restriction estimates and decoupling and to apply them to distance questions. The project is expected to reveal deeper connections between the discrete and continuous setting. An investigation of Fourier restriction for the cone in high dimensions will be continued using algebraic tools in connection with polynomial methods. In addition, the project will explore further applications of Fourier restriction in dispersive partial differential equations.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.