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

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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.