Dp-minimal expansions of $(\mathbb{Z},+)$ via dense pairs via Mordell-Lang

This is a contribution to the classification problem for dp-minimal expansions of $(\mathbb{Z},+)$. Let $S$ be a dense cyclic group order on $(\mathbb{Z},+)$. We use results on "dense pairs" to construct uncountably many dp-minimal expansions of $(\mathbb{Z},+,S)$. These constructions are applications of the Mordell-Lang conjecture and are the first examples of "non-modular" dp-minimal expansions of $(\mathbb{Z},+)$. We canonically associate an o-minimal expansion $\mathcal{R}$ of $(\mathbb{R},+,\times)$, an $\mathcal{R}$-definable circle group $\mathbb{H}$, and a character $\mathbb{Z} \to \mathbb{H}$ to a "non-modular" dp-minimal expansion of $(\mathbb{Z},+,S)$. We also construct a "non-modular" dp-minimal expansion of $(\mathbb{Z},+,\mathrm{Val}_p)$ from the character $\mathbb{Z} \to \mathbb{Z}^\times_p$, $k \mapsto \mathrm{exp}(pk)$.