基于Mordell-Lang的$(\mathbb{Z},+)$稠密对的DP-极小展开
Dp-minimal expansions of $(\mathbb{Z},+)$ via dense pairs via Mordell-Lang
摘要：
这是对$(\mathbb{Z},+)$的dp-极小展开的分类问题的贡献。设$S$是$(\mathbb{Z},+)$上的稠密循环群序.我们使用“密集对”的结果来构造$(\mathbb{Z},+,S)$的不可数的许多dp-极小展开式。这些构造是Mordell-Lang猜想的应用，并且是$(\mathbb{Z},+)$的“非模”dp-极小展开的第一个例子。我们将$(\mathbb{R},+,\times)$的o-最小展开$\mathcal{R}$、$\mathcal{R}$可定义的圆群$\mathbb{H}$和字符$\mathbb{Z} \to \mathbb{H}$与$(\mathbb{Z},+,S)$的“非模”dp-最小展开关联起来。我们还构造了$(\mathbb{Z},+,\mathrm{Val}_p)$的一个“非模”的dp-极小展开式，它由特征标$\mathbb{Z} \to \mathbb{Z}^\times_p$，$k \mapsto \mathrm{exp}(pk)$构成.
英文摘要：
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 $\mathbb{Z} \to \mathbb{Z}^\times_p$, $k \mapsto \mathrm{exp}(pk)$.
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