Distance Questions, Fourier Restriction, and Beyond

距离问题是入射几何的驱动力，入射几何是研究基本几何对象（如点、线或圆）相交模式的数学领域。最著名的距离问题之一叫做Erdös不同距离问题，它要求给定的一组点产生的不同距离的最少个数。另一个例子是单位距离问题，它涉及到一个固定距离在一组给定点之间出现的最大次数。这些问题推动了工具和思想的发展，这些工具和思想在数学以外的学科（如计算机科学、物理学和工程学）中有着广泛的应用。傅里叶限制涉及傅里叶变换，它将一个函数分解成具有不同振荡频率的块。傅里叶限制研究傅里叶变换的一个基本问题：函数的大小与其傅里叶变换的几何关系。这个关系在偏微分方程、数论和其他领域有重要的应用。该研究项目旨在进一步了解距离问题和傅立叶限制之间的相互作用，并开发应用于数学各个领域的现代工具，包括谐波分析，几何测量理论和偏微分方程。该项目的主要方向之一是由Falconer的猜想驱动的，这是对不同距离问题的连续模拟。我们推测当紧集E的Hausdorff维数超过一定阈值时，它的距离集一定有正测度。利用傅里叶限制理论的新思想：入射几何和几何测量理论，得到了解决该猜想的最新结果。本项目将继续研究这些方法，并将其应用于其他相关的距离问题，如一般几何构型、多参数距离和分形测度的投影。该工作旨在进一步研究加权傅里叶限制估计和解耦，并将其应用于距离问题。该项目有望揭示离散和连续设置之间更深层次的联系。高维锥体的傅里叶限制的研究将继续使用代数工具与多项式方法相结合。此外，本项目将进一步探索傅里叶限制在色散偏微分方程中的应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.