权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Research at the Interface of Harmonic Analysis and Arithmetic Combinatorics: Geometric Ramsey Theory and Higher Uniformity Norms

调和分析与算术组合学的接口研究：几何拉姆齐理论和更高的均匀性范数

基本信息

批准号：
1702411
负责人：
Neil Lyall
金额：
$ 15万
依托单位：
University of Georgia Research Foundation Inc
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-15 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1702411&HistoricalAwards=false
关键词：
Research Interface Harmonic Analysis Arithmetic

项目摘要

This project deals with the occurrence of geometric patterns in sufficiently large but otherwise arbitrary sets, a field of mathematics most commonly referred to as (geometric) Ramsey theory. It is specifically focused on the study the question of determining whether or not such sets will be guaranteed to contain a translated and rotated copy of a given finite set, or of its sufficiently large dilates. This study has potential applications to analyzing and determining the true complexity of large data sets. Over the last twenty years, specifically after the groundbreaking work of Gowers on quantitative questions concerning the existence of arbitrarily long sequences of equally spaced numbers in any sufficiently large subsets of the integers, there has been remarkable progress in the study of general linear patterns via the development of so-called higher-order Fourier analysis. Perhaps the most notable achievement here is the celebrated result of Green and Tao on arbitrarily long sequences of equally spaced prime numbers. This project builds on these developments, and one of its major objectives is the development of analytic tools to understand the aforementioned occurrence of geometric and arithmetic structures in large sets. The problems under consideration arise in the context of both the integer lattice and classical Euclidean spaces. The principal investigator's approach blends the well-established delicate interplay between techniques from discrete harmonic analysis and number theory, with a new general approach based on the modern point of view of additive combinatorics.The existence of prescribed geometric structures in large subsets of the integer lattice and also in large measurable subsets of Euclidean spaces is currently not well understood. The project aims to address several such problems using the modern point of view of additive combinatorics. Specifically, the approach of showing that the count of isometric copies of a given finite configuration contained a given set is controlled by certain norms of its so-called balance function. These norms measure the uniformity or randomness of the set and, if it is sufficiently small with respect to the set's density, then the set will contain the expected number of isometric copies. The next step is to establish an inverse theorem showing that the largeness of the norm of the balance function implies that the set correlates or can be approximated by some structured object on which one can iterate this procedure. Recent results of the principal investigator with collaborators indicate that this should indeed the correct framework within which one should be approaching these problems and suggest that questions concerning geometric configurations of different levels of complexity can be tackled in a systematic way using appropriate higher (geometric) uniformity norms. The ultimate goal of this project is to characterizing those finite geometric configuration for which all its sufficiently large dilates can be realized in subsets of Euclidean space of positive upper density, strengthening existing connections between additive combinatorics and classical harmonic analysis, and establishing the analogous characterization in the discrete setting of the integer lattice.

该项目处理足够大但任意的集合中几何图案的出现，这个数学领域通常被称为（几何）拉姆齐理论。它特别关注研究确定是否保证此类集合包含给定有限集合或其足够大的扩张的平移和旋转副本的问题。这项研究对于分析和确定大数据集的真实复杂性具有潜在的应用。在过去的二十年里，特别是在高尔斯在关于任何足够大的整数子集中是否存在任意长的等距数字序列的定量问题上的开创性工作之后，通过所谓的高阶傅里叶分析的发展，一般线性模式的研究取得了显着的进展。也许这里最值得注意的成就是格林和陶在任意长的等距素数序列上取得的著名结果。该项目建立在这些发展的基础上，其主要目标之一是开发分析工具来理解前面提到的大型集合中几何和算术结构的出现。所考虑的问题出现在整数格和经典欧几里得空间的背景下。首席研究员的方法将离散调和分析和数论技术之间已确立的微妙相互作用与基于加法组合学现代观点的新通用方法融合在一起。目前，在整数晶格的大子集中以及欧几里德空间的大可测量子集中，规定的几何结构的存在性还没有得到很好的理解。该项目旨在利用加法组合学的现代观点来解决几个此类问题。具体来说，显示包含给定集合的给定有限配置的等距副本的计数的方法由其所谓的平衡函数的某些规范控制。这些规范测量集合的均匀性或随机性，如果它相对于集合的密度足够小，则集合将包含预期数量的等距副本。下一步是建立一个逆定理，表明平衡函数范数的大意味着该集合与某些结构化对象相关或可以通过某些结构化对象来近似，在该结构化对象上可以迭代此过程。主要研究者与合作者的最新结果表明，这确实应该是解决这些问题的正确框架，并表明可以使用适当的更高（几何）均匀性规范以系统的方式解决有关不同复杂程度的几何配置的问题。该项目的最终目标是表征那些有限几何配置，其所有足够大的扩张都可以在正上密度的欧几里得空间子集中实现，加强加法组合学和经典调和分析之间的现有联系，并在整数晶格的离散设置中建立类似的表征。