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

• Main Author: Hieronymi, Philipp Christian Karl
• Other Authors: Wilkie, A. J.
• Published: University of Oxford 2008
• Subjects: 512.786; Mathematics : Mathematical logic and foundations : O-minimality : Power Function : Model Theory : Diophantine Approximation : Schanuel's Conjecture
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.490071

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 definable set in such a structure is a boolean combination of existentially definable sets and that these structures have o-minimal open core. A proof will be given that the Schanuel conditions used in proving these statements hold for co-countably many real numbers. The results mentioned above will also be established for expansions of R by dense multiplicative subgroups which are closed under all power functions definable in R. In this case the results hold under the assumption that the Conjecture on intersection with tori is true. Finally, the structure consisting of R and the discrete multiplicative subgroup 2^{Z} will be analyzed. It will be shown that this structure is not model complete. Further I develop a connection between the theory of Diophantine approximation and this structure.