Recurrent Ideas in Number Theory: The Multiple Birkhoff Recurrence Theorem Used to Prove van der Waerden's Theorem

Abstract

In 1977 and 1978, some striking and beautiful connections between the fields of topological dynamics and combinatorial number theory began to emerge. Since then there has been an explosion of sensational results that connect these two fields and provide new and clear insights into some classical number-theoretic theorems and combinatorial questions. This story is about classical number-theoretic results being understood in a new and systematic way using the tools and machinery of dynamical systems. The point of this note is to discuss in some detail one such important number-theoretic result and its new dynamical interpretation, namely van der Waerden's theorem, which provides a particularly clear and simple example. (We focus on the dynamical interpretation of this number-theoretic result. Classical proofs of theorems are not included, but the reader is invited to explore the references [6], [71, [2], [8].) Number theory is one of the oldest branches of mathematics. It deals with the natural numbers 1N, which are used for counting and calculations, and it is the solid foundation of modern algebra. Combinatorial number theory, simply speaking, involves questions about sets of natural numbers and classes of sets of natural numbers and the structure of these classes as sets. Questions in combinatorial number theory are quantitative and set-theoretic in nature, as opposed to those concerning algebraic properties of numbers (divisibility, primeness, etc.). In the late 1920's the following problem was posed: If the natural numbers are divided into 2 classes, must there be an arithmetic progression of arbitrary length in one of the classes? Many mathematicians tried to solve this problem-it is appealing due to its (deceptively) simple nature -but the solution by van der Waerden proved to be surprisingly deep. In any event, in 1927 van der Waerden proved the following theorem, answering a slightly more general question.