Structure theorems in tame expansions of o-minimal structures by a dense set

TL;DR

This paper establishes a structure theorem for definable sets and functions in tame expansions of o-minimal structures by a dense set, generalizing cell decomposition and advancing the understanding of definable sets in such structures.

Contribution

It introduces a new structure theorem that decomposes definable sets into unions of cones, extending cell decomposition to tame expansions by dense sets.

Findings

Definable set dimension matches pregeometric dimension and is invariant under definable bijections.

Every definable map is piecewise given by an efinable map outside smaller-dimensional subsets.

Group operation near generic elements is efinable, generalizing known results.

Abstract

We study sets and groups definable in tame expansions of o-minimal structures. Let $\mathcal {\widetilde M}= \langle \mathcal M, P\rangle$ be an expansion of an o-minimal $\mathcal L$-structure $\cal M$ by a dense set $P$, such that three tameness conditions hold. We prove a structure theorem for definable sets and functions in analogy with the influential cell decomposition theorem known for o-minimal structures. The structure theorem advances the state-of-the-art in all known examples of $\mathcal {\widetilde M}$, as it achieves a decomposition of definable sets into \emph{unions} of `cones', instead of only boolean combinations of them. We also develop the right dimension theory in the tame setting. Applications include: (i) the dimension of a definable set coincides with a suitable pregeometric dimension, and it is invariant under definable bijections, (ii) every definable map is given by an $\cal L$-definable map off a subset of its domain of smaller dimension, and (iii) around generic elements of a definable group, the group operation is given by an $\cal L$-definable map.