Poincaré recurrence and number theory: thirty years later

Abstract

Hillel Furstenberg’s 1981 article in the Bulletin gives an elegant introduction to the interplay between dynamics and number theory, summarizing the major developments that occurred in the few years after his landmark paper [21]. The field has evolved over the past thirty years, with major advances on the structural analysis of dynamical systems and new results in combinatorics and number theory. Furstenberg’s article continues to be a beautiful introduction to the subject, drawing together ideas from seemingly distant fields. Furstenberg’s article [21] gave a general correspondence between regularity properties of subsets of the integers and recurrence properties in dynamical systems, now dubbed the Furstenberg Correspondence Principle. He then showed that such recurrence properties always hold, proving what is now referred to as the Multiple Recurrence Theorem. Combined, these results gave a new proof of Szemeredi’s Theorem [45]: if S ⊂ Z has positive upper density, then S contains arbitrarily long arithmetic progressions. This proof lead to an explosion of activity in ergodic theory and topological dynamics, beginning with new proofs of classic results of Ramsey Theory and ultimately leading to significant new combinatorial and number theoretic results. The full implications of these connections have yet to be understood. The approach harks back to the earliest results on recurrence, in the measurable setting and in the topological setting. A measure preserving system is a quadruple (X,B, μ, T ), where X denotes a set, B is a σalgebra on X, μ is a probability measure on (X,B), and T : X → X is a measurable transformation such that μ(T−1(A)) = μ(A) for all A ∈ B. Poincare Recurrence states that if (X,B, μ, T ) is a measure preserving system and A ∈ B with μ(A) > 0, there exists n ∈ N such that μ(A ∩ T−nA) > 0. A (topological) dynamical system is a pair (X,T ), where X is a compact metric space and T : X → X is a continuous map. One can show that any such topological space admits a Borel, probability measure that preserves T . In particular, Poincare Recurrence implies recurrence in the topological setting. Birkhoff [13] gave a direct proof of