The real field with an irrational power function and a dense multiplicative subgroup

@article{Hieronymi2011TheRF,
  title={The real field with an irrational power function and a dense multiplicative subgroup},
  author={Philipp Hieronymi},
  journal={J. Lond. Math. Soc.},
  year={2011},
  volume={83},
  pages={153-167}
}
In recent years the field of real numbers expanded by a multiplicative subgroup has been studied extensively. In this thesis, the known results will be extended to expansions of the real field. I will consider the structure R consisting of the field of real numbers and an irrational power function. Using Schanuel conditions, I will give a first-order axiomatization of expansions of R by a dense multiplicative subgroup which is a subset of the real algebraic numbers. It will be shown that every… Expand
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References

SHOWING 1-10 OF 37 REFERENCES
Dense pairs of o-minimal structures
The structure of definable sets and maps in dense elementary pairs of o-minimal expansions of ordered abelian groups is described. It turns out that a certain notion of “small definable set” plays aExpand
Expansions of the Real Field with Power Functions
  • Chris Miller
  • Mathematics, Computer Science
  • Ann. Pure Appl. Log.
  • 1994
TLDR
It is shown that the (O-minimal) theory of the ordered field of real numbers augmented by all restricted analytic functions and all real power functions admits elimination of quantifiers and has a universal axiomatization. Expand
Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function
Recall that a subset of R is called semi-algebraic if it can be represented as a (finite) boolean combination of sets of the form {~ α ∈ R : p(~ α) = 0}, {~ α ∈ R : q(~ α) > 0} where p(~x), q(~x) areExpand
The model theory of the field of reals with a subgroup of the unit circle
We describe definable sets in the field of reals augmented by a predicate for a finite rank multiplicative group of complex numbers contained in the unit circle . This structure interprets theExpand
Expansions of the real line by open sets: o-minimality and open cores
The open core of a structure R := (R, <, . . .) is defined to be the reduct (in the sense of definability) of R generated by all of its definable open sets. If the open core of R is o-minimal, thenExpand
Elementary properties of ordered abelian groups
Introduction. A complete classification of abelian groups by their elementary properties (i.e. properties that can be formalized in the lower predicate calculus) was given by Szmielew [9]. No suchExpand
A generalization of the Tarski-Seidenberg theorem, and some nondefinability results
This article points out some remarkable facts implicit in the results of Lojasiewicz [LI] and Gabrielov [Ga]. An important consequence of Tarski's work [T] on the elementary theory of the reals is aExpand
DEFINABLE RELATIONS IN THE REAL FIELD WITH A DISTINGUISHED SUBGROUP OF THE UNIT CIRCLE
We describe definable relations in the real field augmented by a binary relation which is an arbitrary multiplicative group of complex numbers contained in the divisible hull of a finitely generatedExpand
Defining the set of integers in expansions of the real field by a closed discrete set
Let D\subseteq \mathbb{R} be closed and discrete and f:D^n \to \mathbb{R} be such that f(D^n) is somewhere dense. We show that (\mathbb{R},+,\cdot,f) defines the set of integers. As an application,Expand
Avoiding the projective hierarchy in expansions of the real field by sequences
Some necessary conditions are given on infinitely oscillating real functions and infinite discrete sets of real numbers so that first-order expansions of the field of real numbers by such functionsExpand
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