On dp-minimal expansions of the integers II

We first prove that if Z is a dp-minimal expansion of (Z,+,0,1) which is not interdefinable with (Z,+,0,1,<), then every infinite subset of Z definable in Z is generic in Z. Using this, we prove that if Z is a dp-minimal expansion of (Z,+,0,1) with monster model G such that G⁰⁰≠ G⁰, then for some α, the cyclic order on Z induced by the embedding n↦ nα+Z of Z in R/Z is definable in Z. The proof employs the Gleason-Yamabe theorem for abelian groups.