The Discrepancy Theory and the Small Ball Conjecture

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The goal of this project is to investigate a number of questions in harmonic analysis, discrepancy theory, approximation theory, and probability as well as connections between these questions. Most of the proposed problems revolve around a harmonic analysis problem -- the Small Ball inequality, which provides a lower bound for hyperbolic sums of Haar functions. While in two dimensions the sharp form of the inequality is established, the higher dimensional case is extremely difficult due to subtle combinatorial complications. This inequality is intimately related to the discrepancy theory, although this connection is yet to be understood. One of the main objects in this theory - the discrepancy function - quantifies the extent of equidistribution of a point set and measures the set's adequacy for numerical integration. It is known that this function must necessarily be large in various senses, however precise estimates in the most interesting cases are established only in two dimensions. The Small Ball inequality is also related to sharp estimates of the small deviation probabilities for Gaussian processes, in particular the Brownian Sheet, and the entropy number estimates of certain mixed derivative spaces in approximation theory. This project investigates important and delicate connections between diverse fields of mathematics. Some parts of the proposed research have potential for applications in areas outside pure mathematics, e.g. mathematical finance. The research proposed in the current project will be actively disseminated through publications, conferences and research visits, leading to important exchange of ideas between mathematicians of different countries, schools, and areas of expertise. The PI also plans to involve students, both graduate and undergraduate, and young mathematicians to participate in this project.