The Discrepancy Theory and the Small Ball Conjecture

差异理论和小球猜想

• 批准号: 0801036
• 负责人: Dmitriy Bilyk
• 金额: $ 10.24万
• 依托单位: University South Carolina Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801036&HistoricalAwards=false
• 关键词: Discrepancy; Theory; Small; Ball; Conjecture

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.

该项目的目标是研究调和分析、差异理论、逼近理论和概率中的一些问题以及这些问题之间的联系。大多数提出的问题都围绕着调和分析问题——小球不等式，它为哈尔函数的双曲和提供了下界。虽然在二维中建立了不等式的尖锐形式，但由于微妙的组合复杂性，高维情况极其困难。这种不平等与差异理论密切相关，尽管这种联系尚待理解。该理论的主要对象之一 - 差异函数 - 量化点集的均匀分布程度并测量该集的数值积分的充分性。众所周知，该函数在各种意义上都必须很大，但是在最有趣的情况下，精确的估计仅在二维上建立。小球不等式还与高斯过程的小偏差概率的尖锐估计有关，特别是布朗表，以及近似理论中某些混合导数空间的熵数估计。 该项目研究不同数学领域之间重要而微妙的联系。拟议研究的某些部分具有在纯数学以外领域应用的潜力，例如数学金融。当前项目提出的研究将通过出版物、会议和研究访问积极传播，从而在不同国家、学校和专业领域的数学家之间进行重要的思想交流。 PI 还计划让研究生和本科生以及年轻数学家参与该项目。