Approximation by $O$-minimal sets in power-bounded $T$-convex valued fields

TL;DR

This paper demonstrates that in certain power-bounded o-minimal T-convex valued fields, definable sets can be approximated by limits of definable sets over the residue field, linking local and global definability.

Contribution

It establishes a precise approximation of definable sets in T-convex valued fields by families of definable sets over the residue field, extending the understanding of definability in power-bounded o-minimal structures.

Findings

Definable sets are limits of families of residue field-definable sets.

Any definable set in an elementary substructure is a trace of a definable set in the larger structure.

The results apply to theories with infinite-dimensional field of exponents.

Abstract

We show that, for a certain large class of power-bounded $o$-minimal $\mathcal{L}_T$-theories $T$ whose field of exponents is infinite-dimensional as a vector space over the rationals, any definable set in a $T$-convex valued field $(\mathcal{R}, \mathfrak{O})$ is in a precise sense the limit of a family of $\mathcal{L}_T$-definable sets indexed over the residue field. Alternatively, in the mainstream model-theoretic language, this says that if $(\mathcal{R}', \mathfrak{O}')$ is an elementary substructure of $(\mathcal{R}, \mathfrak{O})$ and if the residue field of $\mathfrak{O}$ contains an element that is infinitesimal relative to the residue field of $\mathfrak{O}'$ then any set $A \subseteq (\mathcal{R}')^m$ definable in $(\mathcal{R}', \mathfrak{O}')$ is the trace of a set definable in $\mathcal{R}$.