| dc.description.abstract | A set $R \subset{\Z}$ is a \emph{set of recurrence} if, given any invertible measure-preserving transformation $T$ on a probability space $(X, \mathcal{B}, \mu)$ and any set $A \in \mathcal{B}$ with $\mu (A) > 0$, we have $\mu (A\cap T^{-n}A) > 0$ for infinitely many $n \in R$. The study of sets of recurrence is concerned with subsets of ${\N}$ or ${\Z}$ that allow us to predict return to the set $A$ under repeated iterations by a measure-preserving transformation.
In this thesis, we examine measure-preserving systems, ergodicity, the concept of equidistribution, and theorems on recurrence. We give an account of differing and equivalent notions of recurrence with examples. A substantial part of this thesis is dedicated to the formal verification of methods for proving sets of recurrence, followed up by a survey focusing in particular on generalised polynomials. Furthermore, we present Furstenberg's Multiple Recurrence Theorem and a selection of sets of multiple recurrence. Finally, we present Furstenberg's Correspondence Principle, and applications of sets of recurrence in the fields of combinatorics and number theory.
As a novel contribution, we establish recurrence properties for sums of two generalised polynomials, where each consists of a real-valued polynomial vanishing at zero. We achieve this under the assumption that the real-valued polynomials are equidistributed and use a new result by Bergelson, Knutson and Son in the proof. | |