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

这是一个集论、组合学和分析等数学主题为一体的研究项目。该研究包含三个项目，涉及数学的两个主要领域：描述性集合理论和遍历拉姆齐理论。前两个研究项目是可定义等价关系理论，它为理解数学对象的分类本质提供了一个一般框架，直到一些等价的概念；由于其广泛的范围，它与数学的许多领域有自然的相互作用。这两个课题致力于研究可定义等价关系的一个重要子类，以及该子类中成员的微扩展是否仍然属于该子类。第三个项目的特点是用一种新的方法来获得算术组合中的语句，其性质类似于著名的Szemeredi定理，该定理大致表明，任何不可忽略的整数子集保留了整个整数集的大部分加性结构。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，可树等价关系是可数Borel等价关系的一个重要子类。前两个项目在两个不同的背景下研究了这个子类在有限指数扩展下的闭包问题：Borel和测度论。前者涉及Borel组合学和可能的Borel博弈，而后者则与遍历理论和等价关系成本理论紧密相关，并且可能需要几何群论中的非平凡机制。第三个项目是遍历拉姆齐理论，其目标是通过非标准分析提供的对应原理获得可服从群的多次递归结果。这是通过将递归陈述从给定的可服从群转移到更方便的概率群设置来实现的，方法是利用原始群的超功率并为其配备勒布测度。后者是可数加性的，它是概率群相对于原始可服从群的主要优势，而原始可服从群只具有有限可加性的密度函数，使得群上的积分和富比尼定理的使用成为可能。

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.