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

Abstract

We describe definable sets in the field of reals augmented by a predicate for a finite rank multiplicative group Γ of complex numbers contained in the unit circle S. This structure interprets the quotient-space S/Γ which, for Γ 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.

NIP for some pair-like theories

“…Since T is an expansion by constants of the theory of real-closed ordered fields, it is well known that conditions (i) and (iv) are satisfied. We deduce conditions (ii) and (iii) from the results in [1]. The argument overlaps with the reasoning in [1].…”

“…With respect to the result from [6], it has the advantage that it works outside the setting where T is o-minimal. We apply our result to theories of the real field with a predicate for a subgroup of the unit circle which were studied by Belegradek and Zilber in [1]. Similar theories were studied by van den Dries and Günaydın in [5] and by Berenstein, Ealy and Günaydın in [3] and shown to have NIP in [6].…”

“…T Γ was studied by Belegradek and Zilber in [1]. They gave axioms for it and proved a near model completeness result.…”

“…Let T P be the resulting L P -theory of R. Let Γ be a new binary predicate and L Γ = L ∪ {Γ}. Let the suggestively named Γ(R) be the interpretation of Γ in R. Let T Γ be the resulting L Γ -theory of R. As is noted in [1], Γ(R) = Re −1 (Re(Γ(R))) and so T P and T Γ are definitionally equivalent.…”

“…We now consider a class of theories studied by Belegradek and Zilber in [1]. Let R be the real field and S the unit circle thought of as a subgroup of the multiplicative group of the complex field C. Let Γ(R) ≤ S be a subgroup.…”