Koki Okura | Searcharxiv

arXiv2026

Distality in Ordered Abelian Groups

We provide a characterization of distal ordered abelian groups: An ordered abelian group is distal if and only if, for each prime number $p$, the sizes of ribs with respect to the "valuation" $\mathfrak{s}_p$ are uniformly bounded. This generalizes the distality criterion for ordered abelian groups with finite spines given by Aschenbrenner, Chernikov, Gehret, and Ziegler. Furthermore, we prove that every ordered abelian group has a distal expansion, affirmatively answering a conjecture posed by the same authors.

math.LO↗

arXiv2026

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.

math.LO↗