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 $K^m \to \mathbb{Q}^m_p$, and $X \subseteq K^m$ be $K$-definable. Delon has shown that $\mathbb{Q}^m_p \cap X$ is $\mathbb{Q}_p$-definable. Yao has shown that $\dim \mathbb{Q}^m_p \cap X \leq \dim X$ and $\dim \mathrm{st}(V^n \cap X) \leq \dim X$. 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 $\mathbb{Q}^{\mathrm{an}}_p$ of $\mathbb{Q}_p$ by all analytic functions $\mathbb{Z}^n_p \to \mathbb{Q}_p$. As an application we show that if $(X_k)_{k \in \mathbb{N}}$ is a sequence of elements of an $\mathbb{Q}^{\mathrm{an}}_p$-definable family of subsets of $\mathbb{Q}^m_p$ which converges in the Hausdroff topology to $X \subseteq \mathbb{Q}^m_p$ then $X$ is $\mathbb{Q}^{\mathrm{an}}_p$-definable and $\dim X \leq \limsup_{k \to \infty} \dim X_k$.