Pairs of oriented abelian groups

Let $r>1$ be an irrational number and $P_r$ be a predicate standing for the set $\big\{\lfloor nr\rfloor: n\in \Z\setminus \{0\}\big\}$. The first problem we tackled was to find the tameness properties of the expansion $(\Z,+,0,P_r)$ of the group of integers. In \cite{G-O} we give some tameness results of this structure by using the isomorphism between $\Z$ and a subgroup of the unit circle $\mathbb{S}$. Namely, let $h:\Z\rightarrow \Gamma_r$ be the group isomorphism sending an integer $n$ to $\exp (\frac{2\pi i n}{r})$. So we have an isomorphism \beq h: (\Z,+,-,0,1,P_r)\rightarrow \big(\Gamma_r, \cdot,^{-1},1,h(1),h(P_r)\big).\eeq Since $\Gamma_r$ is a subgroup of the unit circle $\mathbb{S}$, it has an orientation on itself. Then $h(P_r)$ becomes an orientation interval. Indeed, the structures $\big(\Gamma_r, \cdot,^{-1},1,h(1),h(P_r)\big)$ and $\big(\Gamma_r, \cdot,\mathcal{O},h(1)\big)$ are interdefinable where $\mathcal{O}$ is the restriction of the orientation of the unit circle $\mathbb{S}$ to $\Gamma_r$. Therefore expanding $\Z$ by $P_r$ is the same as expanding a subgroup of the unit circle by an orientation interval. So it is more convenient to study on these groups called \emph{oriented abelian groups}. First we introduce oriented abelian groups and give the relation with ordered abelian groups. Specifically, we focus on regularly dense oriented abelian groups and give model theoretic results of these groups. We define the $\mathcal{O}$-minimality of oriented abelian groups, which is a similar concept to the o-minimality for ordered abelian groups. Then we show that $\mathcal{O}$-minimality implies NIP. Finally, we give some results about pairs of oriented abelian groups.