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

这个项目处理几何图案在足够大但其他任意集合中的出现，这是一个数学领域，通常被称为（几何）拉姆齐理论。它特别集中于研究确定这样的集合是否将保证包含给定有限集合的平移和旋转副本，或其足够大的扩张的问题。这项研究在分析和确定大型数据集的真正复杂性方面具有潜在的应用价值。在过去的二十年里，特别是在高尔斯（Gowers）关于在任何足够大的整数子集中存在任意长的等间隔数序列的定量问题的开创性工作之后，通过所谓的高阶傅立叶分析的发展，在一般线性模式的研究中取得了显著进展。也许这里最值得注意的成就是格林和陶关于任意长等间隔素数序列的著名结果。这个项目建立在这些发展的基础上，它的主要目标之一是开发分析工具来理解前面提到的几何和算术结构在大集合中的出现。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。首席研究员的方法混合了离散谐波分析和数论技术之间成熟的微妙相互作用，以及基于加法组合学现代观点的新的一般方法。在整数格的大子集和欧几里德空间的大可测子集中，规定几何结构的存在性目前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这样的问题。具体地说，证明包含给定集合的给定有限构型的等距复制数的方法是由其所谓的平衡函数的某些范数控制的。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，则集合将包含期望的等距副本数量。下一步是建立一个反定理，表明平衡函数的范数的大意味着该集合与某个结构化对象相关或可以被某些结构化对象近似，在这些对象上可以迭代该过程。首席研究员和合作者最近的研究结果表明，这确实应该是一个正确的框架，人们应该在其中处理这些问题，并建议有关不同复杂程度的几何构型的问题可以使用适当的更高（几何）均匀性规范以系统的方式解决。本课题的最终目标是在正上密度欧几里得空间的子集上刻画出其所有足够大展开式都可以实现的有限几何构型，加强加性组合学与经典调和分析之间已有的联系，并在整数格的离散设置下建立类似的刻画。

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.