Treeable Equivalence Relations and the Use of Probability Groups in Arithmetic Combinatorics

这是一个集集合论、组合学和分析的数学课题于一身的研究项目。这项研究包括三个项目，涉及两个主要的数学领域：描述集合论和遍历拉姆齐理论。前两个研究项目在于可定义等价关系理论，该理论为理解数学对象分类的性质提供了一个一般框架，直到某种等价概念；由于其范围广泛，它与许多数学领域有着天然的相互作用。这两个项目致力于研究可定义等价关系的一个重要子类，以及这个子类的成员的轻微扩张是否仍然属于它。第三个项目介绍了一种在算术组合学中获得语句的新方法，其性质类似于著名的Szmeredi定理，该定理粗略地指出，任何不可忽略的整数子集都保留了整个整数集的大部分可加结构。在波兰空间上的可定义等价关系理论中，可数Borel等价关系占据了中心位置，可数Borel等价关系是可树等价关系的重要子类。前两个项目在Borel和测度论两个不同的背景下研究了这个子类在有限指数扩张下的闭包问题。前者涉及Borel组合学，也可能涉及Borel对策，而后者与遍历理论和等价关系代价理论密切相关，可能需要几何群论中的非平凡机器。第三个项目是遍历Ramsey理论，它的目标是通过非标准分析提供的对应原理得到顺群的多重递推结果。这是通过将递归语句从给定的服从群转移到更方便的概率群设置来完成的，这是通过利用原始群的超能力并为其配备Loeb度量来实现的。后者是可数可加的，呈现了概率群相对于仅配备有限可加密度函数的原始服从群的主要优势，使得能够在群上积分并使用Fubini定理。

This is a research project at the interface of the mathematical topics of set theory, combinatorics, and analysis. The research contains three projects involving two main areas of mathematics: descriptive set theory and ergodic Ramsey theory. The first two of the research projects lie in the theory of definable equivalence relations, which provides a general framework for understanding the nature of classification of mathematical objects up to some notion of equivalence; due to its broad scope, it has natural interactions with many areas of mathematics. These two projects are devoted to studying an important subclass of definable equivalence relations and whether slight extensions of the members of this subclass still belong to it. The third project features a new method for obtaining statements in arithmetic combinatorics similar in nature to a celebrated theorem of Szemeredi, which roughly states that any non-negligible subset of integers retains much of the additive structure of the entire set of integers.In the theory of definable equivalence relations on Polish spaces, a central place is occupied by countable Borel equivalence relations, an important subclass of which is that of treeable equivalence relations. The first two projects investigate the question of closure of this subclass under finite index extensions in two different contexts: Borel and measure-theoretic. The former involves Borel combinatorics and possibly Borel games, whereas the latter is tightly connected with ergodic theory and the theory of cost of equivalence relations, and may require nontrivial machinery from geometric group theory. The third project lies in ergodic Ramsey theory and its goal is to obtain multiple recurrence results for amenable groups via a correspondence principle provided by nonstandard analysis. This is done by transferring recurrence statements from a given amenable group to a more convenient setting of probability groups by taking the ultrapower of the original group and equipping it with Loeb measure. The latter, being countably additive, presents the main advantage of the probability group over the original amenable group equipped with only a finitely additive density function, enabling integration over the group and the use of Fubini's theorem.