Chromatic Numbers of Cayley Graphs on Z and Recurrence

In 1987 Paul Erdős asked me if the Cayley graph defined on Z by a lacunary sequence has necessarily a finite chromatic number. Below is my answer, delivered to him on the spot but neverpublished, and some additional remarks. The key is the interpretation of the question in terms of return times of dynamical systems. 1 The Cayley graph ZΛ 1.1 Let Λ ⊂ N. By definition, the Cayley graph ZΛ is the graph whose vertices are the integers, and whose edges are the pairs {(n, n + λ) : n ∈ Z, λ ∈ Λ}. The sequence Λ = {λj} is lacunary (with parameter ρ) if λj+1/λj ≥ ρ > 1. The chromatic number χ(Λ) = χ (ZΛ) is the smallest number of colors needed to color ZΛ such that vertices connected by an edge have different colors. For τ ∈ T we denote by ‖τ‖ the distance in T of τ to 0 ∈ T. Theorem 1.1. If Λ is lacunary then χ(Λ) < ∞. The theorem is an immediate corollary of the following theorem: Theorem 1.2. For every ρ > 1 there exists an ε = ε(ρ) > 0 such that for any lacunary Λ with parameter ρ there exist α ∈ T such that ‖λα‖ > ε for all λ ∈ Λ. Proof of Theorem 1.1. Given ρ, divide T into M equal arcs {Ik}, with Mε > 1. Using the α given by Theorem 1.2, set C(n) = j if nα ∈ Ij . QED 1An account did appear recently in chapter 5 of [4].