Nippy proofs of p-adic results of Delon and Yao

Abstract

Let $K$ be an elementary extension of ${\mathbb{Q}}_p$, $V$ be the set of finite $a\in K$, $\mathrm{st}$ be the standard part map $K^m\to {\mathbb{Q}}_p^m$, and $X\subseteq K^m$ be $K$-definable. Delon has shown that ${\mathbb{Q}}_p^m\cap X$ is ${\mathbb{Q}}_p$-definable. Yao has shown that $\dim {\mathbb{Q}}_p^m\cap X\le \dim X$ and $\dim \mathrm{st}(V^n\cap X)\le \dim X$. We give new $\mathrm{NIP}$-theoretic proofs of these results and show that both inequalities hold in much more general settings. We also prove the analogous results for the expansion ${\mathbb{Q}}_p^{\mathrm{an}}$ of ${\mathbb{Q}}_p$ by all analytic functions ${\mathbb{Z}}_p^n\to {\mathbb{Q}}_p$. As an application we show that if $(X_k{)}_{k\in \mathbb{N}}$ is a sequence of elements of an ${\mathbb{Q}}_p^{\mathrm{an}}$-definable family of subsets of ${\mathbb{Q}}_p^m$ which converges in the Hausdroff topology to $X\subseteq {\mathbb{Q}}_p^m$ then $X$ is ${\mathbb{Q}}_p^{\mathrm{an}}$-definable and $\dim X\le {\mathrm{limsup}}_{k\to \infty }\dim X_k$.

preliminary version, comments are welcome

References in corpus (1)

• On Dimensions, Standard Part Maps, and $p$-Adically Closed Fields