Let ℜ be an o-minimal expansion of (ℝ, <+) and (φk)k∈ℕ be a sequence of positive real numbers such that limk→+∞f(φk)/φk+1=0 for every f:ℝ→ ℝ definable in ℜ. (Such sequences always exist under some reasonable extra assumptions on ℜ, in particular, if ℜ is exponentially bounded or if the language is countable.) Then (ℜ, (S)) is d-minimal, where S ranges over all subsets of cartesian powers of the range of φ.

We show that if is not in the field generated by α1,…,αn, then no restriction of the function xβ to an interval is definable in . We also prove that if the real and imaginary parts of a complex analytic function are definable in Rexp or in the expansion of by functions xα, for irrational α, then they are already definable in . We conclude with some conjectures and open questions.

We develop a general framework for measure theory and integration theory that is compatible with o-minimality. Therefore the following natural definitions are introduced. Given are an o-minimal structure M and a Borel measure μ on some Rn. We say that μ is M-compatible if there is an o-minimal expansion of M such that for every parameterized family of subsets of Rn that is definable in M the corresponding family of μ-measures is definable in this o-minimal expansion. We say that μ (...) is M-tame if there is an o-minimal expansion of M such that for every parameterized family of functions on Rn that is definable in M the corresponding family of integrals with respect to μ is definable in this o-minimal expansion. We substantiate these definitions with existing and many new examples. We investigate the Lebesgue measure in their light. We prove definable versions of fundamental results such as the theorem of Radon–Nikodym, the Lebesgue decomposition theorem and the Riesz representation theorem. They allow us to describe explicitly compatible and tame measures and to classify them in dimension one. (shrink)

We prove that given any first order formula φ in the language L' = {+,., <, (f i ) i ∈ I , (c i ) i ∈ I }, where the f i are unary function symbols and the c i are constants, one can find an existential formula ψ such that φ and ψ are equivalent in any L'-structure $\langle {\Bbb R},+,.,<,(x^{c_{i}})_{i\in I},(c_{i})_{i\in I}\rangle $.

We show that the theory of the real field with a generic real power function is decidable, relative to an oracle for the rational cut of the exponent of the power function. We also show the existence of generic computable real numbers, hence providing an example of a decidable o-minimal proper expansion of the real field by an analytic function.

Let T be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of T and show that the residue field of such a convex hull has a natural expansion to a model of T. We give a quantifier elimination relative to T for the theory of pairs (R, V) where $\mathscr{R} \models T$ and V ≠ R is the convex hull of an elementary substructure of R. We deduce (...) that the theory of such pairs is complete and weakly o-minimal. We also give a quantifier elimination relative to T for the theory of pairs (R, N) with R a model of T and N a proper elementary substructure that is Dedekind complete in R. We deduce that the theory of such "tame" pairs is complete. (shrink)

We introduce the set of definable restricted complex powers for expansions of the real field and calculate it explicitly for expansions of the real field itself by collections of restricted complex powers. We apply this computation to establish a classification theorem for expansions of the real field by families of locally closed trajectories of linear vector fields.

We study the Hardy field associated with an o-minimal expansion of the real numbers. If the set of analytic germs is dense in the Hardy field, then we can definably analytically separate sets in , and we can definably analytically approximate definable continuous unary functions. A similar statement holds for definable smooth functions.

We give necessary and sufficient conditions on a non-oscillatory curve in an o-minimal field such that, for any bounded definable function, the germ of the function on an initial segment of the curve has a definable extension to a closed set. This situation is translated into a question about types: What are the conditions on an n-type such that, for any bounded definable function, the germ of the function on the type has a definable continuous global extension? Certain categories of (...) definable types have this property, and we give the precise conditions that are equivalent to existence of the global extension. (shrink)