Uniform Distribution and Harmonic Analysis

This research project deals with the fundamental problem of approximating continuous objects by discrete ones and evaluating the errors that inevitably arise in the process. Questions of this kind come up in most branches of mathematics: probability, approximation theory, number theory, combinatorics, and discrete geometry, as well as in any discipline of science, engineering, or finance that demands computations of multivariate integrals. A number of less obvious and more profound connections have been discovered recently: some have already been formalized and understood, while others are only heuristic and await further research. Almost every pivotal development in uniform distribution theory, the focus of this project, has exploited applications of real analysis, functional analysis, and especially harmonic analysis. At the same time, numerous modern methods of harmonic analysis that were historically overlooked by experts in other fields that make use of uniform distribution theory are only starting to gain footholds in their areas.

Many questions of utmost importance to uniform distribution theory, especially in higher dimensions, remain wide open. The questions under investigation in this project include the longstanding problem of precise asymptotics of optimal discrepancy in higher dimensions, one-bit compressed sensing, embeddings of metric spaces, interplay between discrepancy and energy minimization, constructions of well-distributed point sets (low-discrepancy, cubature, energy-minimizing, lattices), discrepancy estimates and numerical integration in function spaces, exploring the effect of geometry on uniform distribution properties, and problems of discrete geometry (tessellations, coverings, packings). Progress on these questions has to be tightly intertwined with advances on important problems and conjectures in analysis and other fields, gluing together a mosaic of apparently disconnected questions and topics. The outcomes of the project are expected to impact several areas of mathematics, enriching and cross-fertilizing them with new results, ideas, and methods.