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.

这个项目的目标是调查谐波分析，差异理论，近似理论和概率以及这些问题之间的联系的一些问题。大多数提出的问题围绕着调和分析问题-小球不等式，它提供了一个下界的双曲和哈尔函数。虽然在二维的尖锐形式的不等式成立，高维的情况下是非常困难的，由于微妙的组合的复杂性。这种不平等与差异理论密切相关，尽管这种联系尚待理解。在这个理论的主要对象之一-差异功能-量化的程度均匀分布的点集和措施集的充分数值积分。众所周知，这个函数在各种意义上都必须很大，但是在最有趣的情况下，精确的估计只能在两个维度上建立。小球不等式也与高斯过程的小偏差概率的精确估计有关，特别是布朗单，以及近似理论中某些混合导数空间的熵数估计。 该项目研究数学的不同领域之间的重要和微妙的联系。某些部分的拟议研究有潜力的应用领域以外的纯数学，如数学金融。本项目中提出的研究将通过出版物、会议和研究访问积极传播，导致不同国家、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的数学家参与这个项目。