On continuous functions definable in expansions of the ordered real additive group

Let $\mathcal R$ be an expansion of the ordered real additive group. Then one of the following holds: either every continuous function $[0,1] \to \mathbb R$ definable in $\mathcal R$ is $C^2$ on an open dense subset of $[0,1]$, or every $C^2$ function $[0,1] \to \mathbb R$ definable in $\mathcal R$ is affine, or every continuous function $[0,1] \to \mathbb R$ is definable in $\mathcal R$. If $\mathcal R$ is NTP$_{2}$ or more generally does not interpret a model of the monadic second order theory of one successor, the first case holds. It is due to Marker, Peterzil, and Pillay that whenever $\mathcal R$ defines a $C^2$ function $[0,1] \to \mathbb R$ that is not affine, it also defines an ordered field on some open interval whose ordering coincides with the usual ordering on $\mathbb R$. Assuming $\mathcal R$ does not interpret second-order arithmetic, we show that the last statement holds when $C^2$ is replaced by $C^1$

Tame Topology over Dp-minimal Structures

We develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets are tame: there is a good notion of dimension on definable sets, definable functions are almost everywhere continuous, and definable sets are finite unions of graphs of definable continuous "multi-valued functions". This generalizes known statements about weakly o-minimal, C-minimal and P-minimal theories.Comment: 14 pages, some minor errors fixe

Interpreting the monadic second order theory of one successor in expansions of the real line

We give sufficient conditions for a first order expansion of the real line to define the standard model of the monadic second order theory of one successor. Such an expansion does not satisfy any of the combinatorial tameness properties defined by Shelah, such as $\textrm{NIP}$ or even $\textrm{NTP}_2$. We use this to deduce the first general results about definable sets in $\textrm{NTP}_2$ expansions of $(\mathbb{R},<,+)$

The Hausdorff dimension of metric spaces definable in o-minimal expansions of the real field

Let $R$ be an o-minimal expansion of the real field. We show that the Hausdorff dimension of an $R$-definable metric space is an $R$-definable function of the parameters defining the metric space. We also show that the Hausdorff dimension of an $R$-definable metric space is an element of the field of powers of $R$. The proof uses a basic topological dichotomy for definable metric spaces due to the second author, and the work of the first author and Shiota on measure theory over nonarchimedean o-minimal structures

A family of dp-minimal expansions of (Z;+)(\mathbb{Z};+)

We show that the cyclically ordered-abelian groups expanding $(\mathbb{Z};+)$ contain a continuum-size family of dp-minimal structures such that no two members define the same subsets of $\mathbb{Z}$

Dp-minimal expansions of discrete ordered abelian groups

If $\mathcal{Z}$ is a dp-minimal expansion of a discrete ordered abelian group $(Z,<,+)$ and $\mathcal{Z}$ does not admit a nontrivial definable convex subgroup then $\mathcal{Z}$ is interdefinable with $(Z,<,+)$ and $(Z,<,+)$ is elementarily equivalent to $(\mathbb{Z},<,+)$.Comment: This has be superseded by arXiv:1909.0539

Notes on trace equivalence

We introduce trace definability, a weak notion of interpretability, and trace equivalence, a weak notion of equivalence for first order structures and theories. In particular we get an interesting weak equivalence notion for $\mathrm{NIP}$ theories. We describe a close connection to indiscernible collapse. We also show that if $Q$ is a divisible subgroup of $(\mathbb{R};+)$ and $\mathcal{Q}$ is a dp-rank one expansion of $(Q;+,<)$ then exactly one of the following holds: $\mathrm{Th}(\mathcal{Q})$ trace defines $\mathrm{RCF}$ or $\mathcal{Q}$ is trace equivalent to a reduct of an ordered vector space.Comment: This supersedes and replaces arxiv:1910.13504 and arxiv:2006.0013

Dp-minimal valued fields

We show that dp-minimal valued fields are henselian and that a dp-minimal field admitting a definable type V topology is either real closed, algebraically closed or admits a non-trivial definable henselian valuation. We give classifications of dp-minimal ordered abelian groups and dp-minimal ordered fields without additional structure

An NIP structure which does not interpret an infinite group but whose Shelah expansion interprets an infinite field

We describe one.Comment: 3 page

Wild theories with o-minimal open core

Let $T$ be a consistent o-minimal theory extending the theory of densely ordered groups and let $T'$ be a consistent theory. Then there is a complete theory $T^*$ extending $T$ such that $T$ is an open core of $T^*$, but every model of $T^*$ interprets a model of $T'$. If $T'$ is NIP, $T^*$ can be chosen to be NIP as well. From this we deduce the existence of an NIP expansion of the real field that has no distal expansion