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

这个项目涉及在足够大但任意的集合中出现的几何图案，这是一个数学领域，通常被称为（几何）拉姆齐理论。它是专门集中在研究的问题，确定是否或不这样的集合将保证包含一个翻译和旋转的副本，一个给定的有限集，或其足够大的膨胀。该研究对分析和确定大型数据集的真实复杂性具有潜在的应用价值。在过去的20年里，特别是在高尔斯开创性地研究了关于在任何足够大的整数子集中存在任意长的等距数列的定量问题之后，通过发展所谓的高阶傅立叶分析，在一般线性模式的研究中取得了显着的进展。也许最显着的成就在这里是著名的结果绿色和陶对任意长序列的等距素数。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，以了解前面提到的大集合中几何和算术结构的发生。所考虑的问题出现在整数格和经典欧氏空间的背景下。首席研究员的方法融合了离散调和分析和数论技术之间的良好建立的微妙相互作用，与一个新的一般方法的基础上的现代观点的加法组合学。存在规定的几何结构在大子集的整数格，也在大的可测量子集的欧几里得空间目前还没有得到很好的理解。该项目的目的是解决几个这样的问题，使用现代的观点添加剂组合。具体地说，方法表明，一个给定的有限配置包含一个给定的集合的等距副本的计数是由其所谓的平衡函数的某些范数控制。这些范数度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，那么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，证明平衡函数的范数的大意味着集合与某种结构化的对象相关或可以近似，在这种结构化的对象上可以执行这个过程。最近的主要研究者与合作者的结果表明，这确实应该是正确的框架内，人们应该接近这些问题，并建议有关的几何配置的不同层次的复杂性的问题，可以解决在一个系统的方式使用适当的更高的（几何）均匀性规范。 本项目的最终目标是刻画那些有限的几何构型，对于这些构型，其所有足够大的膨胀都可以在正上密度的欧几里得空间的子集中实现，加强加法组合学与经典调和分析之间的现有联系，并在整数格的离散设置中建立类似的刻画。

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.