Distance Graphs with Finite Chromatic Number

The distance graph G(D) with distance set D={d1, d2, …} has the set Z of integers as vertex set, with two vertices i, j∈Z adjacent if and only if |i−j|∈D. We prove that the chromatic number of G(D) is finite whenever inf{di+1/di}>1 and that every growth speed smaller than this admits a distance set D with infinite-chromatic G(D).