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.
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)
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.