Small sets in dense pairs

Let $\widetilde{\mathcal M}=\langle \mathcal M, P\rangle$ be an expansion of an o-minimal structure $\mathcal M$ by a dense set $P\subseteq M$, such that three tameness conditions hold. We prove that the induced structure on $P$ by $\mathcal M$ eliminates imaginaries. As a corollary, we obtain that every small set $X$ definable in $\widetilde{\mathcal M}$ can be definably embedded into some $P^l$, uniformly in parameters, settling a question from [10]. We verify the tameness conditions in three examples: dense pairs of real closed fields, expansions of $\mathcal M$ by a dense independent set, and expansions by a dense divisible multiplicative group with the Mann property. Along the way, we point out a gap in the proof of a relevant elimination of imaginaries result in Wencel [17]. The above results are in contrast to recent literature, as it is known in general that $\widetilde{\mathcal M}$ does not eliminate imaginaries, and neither it nor the induced structure on $P$ admits definable Skolem functions

Characterizing o-minimal groups in tame expansions of o-minimal structures

We establish the first global results for groups definable in tame expansions of o-minimal structures. Let $\mathcal N$ be an expansion of an o-minimal structure $\mathcal M$ that admits a good dimension theory. The setting includes dense pairs of o-minimal structures, expansions of $\mathcal M$ by a Mann group, or by a subgroup of an elliptic curve, or a dense independent set. We prove: (1) a Weil's group chunk theorem that guarantees a definable group with an o-minimal group chunk is o-minimal, (2) a full characterization of those definable groups that are o-minimal as those groups that have maximal dimension; namely their dimension equals the dimension of their topological closure, (3) if $\mathcal N$ expands $\mathcal M$ by a dense independent set, then every definable group is o-minimal

Counting algebraic points in expansions of o-minimal structures by a dense set

The Pila-Wilkie theorem states that if a set $X\subseteq \mathbb R^n$ is definable in an o-minimal structure $\mathcal R$ and contains `many' rational points, then it contains an infinite semialgebraic set. In this paper, we extend this theorem to an expansion $\widetilde{\mathcal R}=\langle \mathcal R, P\rangle$ of $\mathcal R$ by a dense set $P$, which is either an elementary substructure of $\mathcal R$, or it is independent, as follows. If $X$ is definable in $\widetilde{\mathcal R}$ and contains many rational points, then it is dense in an infinite semialgebraic set. Moreover, it contains an infinite set which is $\emptyset$-definable in $\langle \overline{\mathbb R}, P\rangle$, where $\overline {\mathbb R}$ is the real field

Product cones in dense pairs

Let $\mathcal M=\langle M, <, +, \dots\rangle$ be an o-minimal expansion of an ordered group, and $P\subseteq M$ a dense set such that certain tameness conditions hold. We introduce the notion of a `product cone' in $\widetilde{\mathcal M}=\langle \cal M, P\rangle$, and prove: if $\mathcal M$ expands a real closed field, then $\widetilde{\mathcal M}$ admits a product cone decomposition. If $\mathcal M$ is linear, then it does not. In particular, we settle a question from [10]

Semi-linear stars are contractible

Let $\cal R$ be an ordered vector space over an ordered division ring. We prove that every definable set $X$ is a finite union of relatively open definable subsets which are definably simply-connected, settling a conjecture from [5]. The proof goes through the stronger statement that the star of a cell in a special linear decomposition of $X$ is definably simply-connected. In fact, if the star is bounded, then it is definably contractible

Definable quotients of locally definable groups

We study locally definable abelian groups \CU in various settings and examine conditions under which the quotient of \CU by a discrete subgroup might be definable. This turns out to be related to the existence of the type-definable subgroup \CU^{00} and to the divisibility of \CU

Strongly minimal groups in o-minimal structures

We prove Zilber's Trichotomy Conjecture for strongly minimal expansions of two-dimensional groups, definable in o-minimal structures: Theorem. Let M be an o-minimal expansion of a real closed field, (G;+) a 2-dimensional group definable in M, and D = (G;+,...) a strongly minimal structure, all of whose atomic relations are definable in M. If D is not locally modular, then an algebraically closed field K is interpretable in D, and the group G, with all its induced D-structure, is definably isomorphic in D to an algebraic K-group with all its induced K-structure

Small sets in Mann pairs

Let $\widetilde{\mathcal M}=\langle \mathcal M, G\rangle$ be an expansion of a real closed field $\mathcal M$ by a dense subgroup $G$ of $\langle M^{>0}, \cdot\rangle$ with the Mann property. We prove that the induced structure on $G$ by $\mathcal M$ eliminates imaginaries. As a consequence, every small set $X$ definable in $\mathcal M$ can be definably embedded into some $G^l$, uniformly in parameters. These results are proved in a more general setting, where $\widetilde{\mathcal M}=\langle \mathcal M, P\rangle$ is an expansion of an o-minimal structure $\mathcal M$ by a dense set $P\subseteq M$, satisfying three tameness conditions

On definable Skolem functions in weakly o-minimal non-valuational structures

We prove that all known examples of weakly o-minimal non-valuational structures have no definable Skolem functions. We show, however, that such structures eliminate imaginaries up to (definable families of) definable cuts. Along the way we give some new examples of weakly o-minimal non-valuational structures

Locally definable subgroups of semialgebraic groups

We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic subset of $G$. Then the group generated by $X$ contains a generic set and, if connected, it is divisible. More generally, the same result holds when $X$ is definable in any o-minimal expansion of $R$ which is elementarily equivalent to $\mathbb R_{an,exp}$. We observe that the above statement is equivalent to saying: there exists an $m$ such that $\Sigma_{i=1}^m(X-X)$ is an approximate subgroup of $G$.Comment: Small changes in the body of the text with respect to the previous version. The title has changed. The appendix has been shortene