We prove that certain pairs of ordered structures are dependent. Among these structures are dense and tame pairs of o-minimal structures and further the real field with a multiplicative subgroup with the Mann property.
We give a criterion when an expansion of the ordered set of real numbers defines the image of (R, +, ·, N) under a semialgebraic injection. In particular, we show that for a non-quadratic irrational number α, the expansion of the ordered Q(α)-vector space of real numbers by N defines multiplication on R.
Every definably complete expansion of an ordered field satisfies an analogue of the Baire Category Theorem.
The theory of (R, <, +, Z, Za) is decidable if a is quadratic. If a is the golden ratio, (R, <, +, Z, Za) defines multiplication by a. The results are established by using the Ostrowski numeration system based on the continued fraction expansion of a to define the above structures in monadic second order logic of one successor. The converse that (R, <, +,… (More)
For first-order expansions of the field of real numbers, nondefinability of the set of natural numbers is equivalent to equality of topological and Assouad dimension on images of closed definable sets under definable continuous maps.
A Cantor set is a non-empty, compact set that has neither interior nor isolated points. In this paper a Cantor set K ⊆ R is constructed such that every set definable in (R, <, +, ·, K) is Borel. In addition, we prove quantifier-elimination and completeness results for (R, <, +, ·, K), making the set K the first example of a modeltheoretically tame Cantor… (More)
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 NIP or even NTP 2. We use this to deduce the first general results about definable sets in NTP… (More)
The aim of this note is to determine whether certain non-o-minimal expansions of o-minimal theories which are known to be NIP, are also distal. We observe that while tame pairs of o-minimal structures and the real field with a discrete multiplicative subgroup have distal theories, dense pairs of o-minimal structures and related examples do not.
An expansion of a definably complete field either defines a discrete subring, or the image of every definable discrete set under every definable map is nowhere dense. As an application we show a definable version of Lebesgue's differentiation theorem.