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Journal of the London Mathematical Society Advance Access published online on July 10, 2008
Journal of the London Mathematical Society, doi:10.1112/jlms/jdn037
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© 2008 London Mathematical Society

The model theory of the field of reals with a subgroup of the unit circle

Oleg Belegradek

Department of Mathematics
Istanbul Bilgi University
80370 Dolapdere--Istanbul
Turkey
olegb@bilgi.edu.tr

Boris Zilber

Mathematical Institute
University of Oxford
24–29 St Giles
Oxford
OX1 3LB
United Kingdom

We describe definable sets in the field of reals augmented by a predicate for a finite rank multiplicative group {Gamma} of complex numbers contained in the unit circle S. This structure interprets the quotient-space S/{Gamma} which, for {Gamma} infinite cyclic, is related to the quantum torus. Every definable set is proved to be a Boolean combination of existentially definable sets. We give a complete set of axioms for the theory of such a structure.

2000 Mathematics Subject Classification 03C60 (primary), 03C10, 03C64, 11U09 (secondary).

The authors thank the Isaac Newton Institute for Mathematical Sciences, where they obtained the results of the paper in January–February, 2005.

Received June 19, 2007; revised April 8, 2008;
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