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

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.