整数的远距离扩展与p-adic域
Distal Expansions of the Integers and the $p$-adic Fields
AI总结：
本文研究了通过单射投影映射产生的子集对远距离结构的扩展，证明了三种扩展保持远距离性质，包括整数和p-adic域的扩展。
摘要：
本文研究了通过单射投影映射产生的子集对远距离结构的扩展。我们首先提供了一个充分条件，使此类扩展保持远距离。基于此准则，我们建立了三种涉及整数或p-adic域的扩展的远距离性。设R为几乎稀疏序列。我们证明了(Z;<,+;R)是远距离的，从而回答了Tong提出的问题。此外，我们展示了(Q_p;+,*;p^Z)和(Q_p;+,*;p^Z;p^R)的远距离性。后者提供了一个没有Poincaré级数有理性的NIP扩展例子。
英文摘要：
This paper investigates expansions of distal structures by a unary subset that arises as the image of a projection map. We first provide a sufficient condition for such an expansion to remain distal. Based on this criterion, we establish the distality of three kinds of expansions involving the integers or the $p$-adic fields. Let $R$ be an almost sparse sequence. We prove that $(\mathbb{Z};<,+,R)$ is distal, thereby answering a question posed by Tong. Furthermore, we show the distality of $(\mathbb{Q}_p;+,\cdot,p^{\mathbb{Z}})$ and $(\mathbb{Q}_p;+,\cdot,p^{\mathbb{Z}},p^R)$.The latter provides an example of a NIP expansion of the $p$-adic field without the rationality of the Poincaré series.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
