Distal Expansions of the Integers and the $p$-adic Fields

TL;DR

This paper explores conditions under which expansions of distal structures, including integers and p-adic fields, remain distal, providing new examples and answering existing questions in model theory.

Contribution

It establishes a sufficient condition for distal expansions and proves the distality of specific expansions involving integers and p-adic fields, including new examples.

Findings

$( Z; <,+,R)$ is distal for almost sparse sequences R

$( Q_p; +,or p^{ Z}}$ is distal

$( Q_p; +,or p^{ Z}},p^R)$ is distal and provides a non-rational Poincare9 series example

Abstract

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 Poincar\'e series.