Lipschitz extensions of definable p-adic functions

In this paper, we prove a definable version of Kirszbraun's theorem in a non-Archimedean setting for definable families of functions in one variable. More precisely, we prove that every definable function $f : X \times Y \to \mathbb{Q}_p^s$, where $X\subset \mathbb{Q}_p$ and $Y \subset \mathbb{Q}_p^r$, that is $\lambda$-Lipschitz in the first variable, extends to a definable function $\tilde{f}:\mathbb{Q}_p\times Y \to \mathbb{Q}_p^s$ that is $\lambda$-Lipschitz in the first variable.Comment: 11 page

Cell decomposition and classification of definable sets in p-optimal fields

We prove that for p-optimal fields (a very large subclass of p-minimal fields containing all the known examples) a cell decomposition theorem follows from methods going back to Denef’s paper [7]. We derive from it the existence of definable Skolem functions and strong p-minimality. Then we turn to strongly p-minimal fields satisfying the Extreme Value Property—a property which in particular holds in fields which are elementarily equivalent to a p-adic one. For such fields K, we prove that every definable subset of K × K d whose fibers over K are inverse images by the valuation of subsets of the value group is semialgebraic. Combining the two we get a preparation theorem for definable functions on p-optimal fields satisfying the Extreme Value Property, from which it follows that infinite sets definable over such fields are in definable bijection iff they have the same dimension

On non-compact pp-adic definable groups

Peterzil and Steinhorn proved that if a group $G$ definable in an $o$-minimal structure is not definably compact, then $G$ contains a definable torsion-free subgroup of dimension one. We prove here a $p$-adic analogue of the Peterzil-Steinhorn theorem, in the special case of abelian groups. Let $G$ be an abelian group definable in a $p$-adically closed field $M$. If $G$ is not definably compact then there is a definable subgroup $H$ of dimension one which is not definably compact. In a future paper we will generalize this to non-abelian $G$.Comment: 24 page