Weakly minimal groups with a new predicate

Abstract

Fix a weakly minimal (i.e. superstable [Formula: see text]-rank [Formula: see text]) structure [Formula: see text]. Let [Formula: see text] be an expansion by constants for an elementary substructure, and let [Formula: see text] be an arbitrary subset of the universe [Formula: see text]. We show that all formulas in the expansion [Formula: see text] are equivalent to bounded formulas, and so [Formula: see text] is stable (or NIP) if and only if the [Formula: see text]-induced structure [Formula: see text] on [Formula: see text] is stable (or NIP). We then restrict to the case that [Formula: see text] is a pure abelian group with a weakly minimal theory, and [Formula: see text] is mutually algebraic (equivalently, weakly minimal with trivial forking). This setting encompasses most of the recent research on stable expansions of [Formula: see text]. Using various characterizations of mutual algebraicity, we give new examples of stable structures of the form [Formula: see text]. Most notably, we show that if [Formula: see text] is a weakly minimal additive subgroup of the algebraic numbers, [Formula: see text] is enumerated by a homogeneous linear recurrence relation with algebraic coefficients, and no repeated root of the characteristic polynomial of [Formula: see text] is a root of unity, then [Formula: see text] is superstable for any [Formula: see text].