Defining Additive Subgroups of the Reals from Convex Subsets

Let G be a subgroup of the additive group of real numbers and let C ⊆ G be infinite and convex in G. We show that G is definable in (R,+, ⋅, C) and that Z is definable if G has finite rank. This has a number of consequences for expansions of certain o-minimal structures on the real field by multiplicative groups of complex numbers. We answer some questions about expansions of o-minimal structures on the real field R := (R,+, ⋅ ) by various groups. For more detailed treatment of some of the issues raised below, see van den Dries and Günaydın [4] and Miller [6, 7]. In this paper, definable means first-order definable with parameters from the appropriate underlying set. Throughout, let K (= (K,+, ⋅, < )) be an ordered subfield of R, and let G be a nontrivial additive subgroup of K. Proposition 1. Let C ⊆ G be infinite and convex (in G). Then G is definable in (K,C). Proof. If C is unbounded, then G = C ∪ (−C) ∪ (a+C) for some a ∈ K. Suppose now that C is bounded. Then G is not discrete and is thus dense in K. By translation and division by some nonzero element of G, we reduce to the case that 1 ∈ G and C = G ∩ [−1, 1]. Put S = { s ∈ K : ∃ ε > 0, sC ∩ [0, ε) ⊆ C }. Evidently, (K,C) defines S, and S is a subgroup of (K,+) containing 1. Hence, it suffices to show that S ∩ (0, 1) ⊆ C, for then G = S + C. Let s ∈ S ∩ (0, 1) and ε > 0 be such that s ( C ∩ [0, ε) ⊆ C. By density of G, there exists c ∈ C ∩ (0, ε). Note that sc ∈ C ⊆ G. Let k be the smallest positive integer such that kc > 1− ε. Then skc ∈ G and 1 − kc ∈ C ∩ (−ε, ε), so that s(1 − kc) ∈ G as well. Since s = s(1− kc) + skc, we have s ∈ G as required. □ Proposition 2. If G has finite rank (equivalently, if the Q-linear span of G is finite dimensional) and C ⊆ G is infinite and convex, then (K,C) defines Z. Proof. By Proposition 1 and division by some nonzero element of G, we reduce to the case that C = G and 1 ∈ G. Put R = { r ∈ K : rG ⊆ G }. Observe that R is a subring of K contained in G and is definable in (K,G). Since R ⊆ G, it has finite rank as an additive group. Hence, the fraction field J of R is a finite-degree algebraic extension of Q. By J. Robinson [10], Z is definable in (J,+, ⋅). Since J is definable in (K,R), so is Z. □ Received by the editors October 1, 2008, and, in revised form, December 22, 2008, and February 14, 2009. 2000 Mathematics Subject Classification. Primary 03C64; Secondary 14P10. c ⃝2009 American Mathematical Society Reverts to public domain 28 years from publication 3473 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 3474 M. TYCHONIEVICH Remarks. (i) If G is finitely generated and 1 ∈ G, then (K,G) defines Z for all subrings K of R containing G [6, 6.1]. (ii) The set R contains S as in the proof of Proposition 1; equality holds if and only if G is dense. (iii) Of course, undecidability of Th(K,C) follows, but much more is true when K = R: every real Borel set is definable in (R,Z). (iv) Any conditions on G that force the field J to be either a finite-degree algebraic extension of Q or a purely transcendental extension of some J ′ ⊆ J yield definability of Z, again by [10] for the former and by R. Robinson [11] for the latter. A more recent result of Poonen [9] shows that it is sufficient for J to be a finitely generated field extension of Q. (v) If K is real closed and G is a real-closed subfield of K, then (K,G) does not define Z by van den Dries [3]. We do not know of any G other than real-closed subfields that do not define Z over R. We now concentrate on the case that K = R and consider expansions of R by multiplicative subgroups of C∖{0}. We make the usual identifications R ↪→ C ∼= R. Given Z ⊆ C, let e denote the image of Z under complex exponentiation. We focus here on expansions of structures of the form (R, (e)w∈W ) where W ⊆ C. At present, there are essentially only two positive results known: ∙ [6, 7]. If G is cyclic and R is an o-minimal expansion of R that defines no irrational power functions, then (R, e) does not define Z. Indeed, every definable set (of any arity) is a boolean combination of open sets (but much more is true). Write G = αZ, put ω = 2π/α, and suppose moreover that R defines the restriction of the function t → sin(ω log t) : R → R to some nontrivial subinterval of R. Then (R, e) defines the group e. By [2], the expansion of R by the restriction sin(ω log t)↾[1, 2] is o-minimal and defines no irrational power functions. This means that (R, e) does not define Z. ∙ [4]. If G is noncyclic and has finite rank, then neither (R, e) nor (R, e) defines Z. Indeed, every definable set (of any arity) is a boolean combination of Fσ sets. Let Re and Rs denote the expansions of R by the restrictions exp ↾[0, 1] and sin ↾[0, 2π] of the real exponential and sine functions. The intervals of definition are picked for convenience: by addition formulas, for any bounded [a, b] ⊆ R we have exp ↾[a, b] definable in Re and sin ↾[a, b] definable in Rs. By [2], both Re and Rs are o-minimal and define no irrational power functions (indeed, the same is true of their amalgam). It is worth noting that, by Bianconi [1], Rs does not define exp ↾[0, 1] and Re does not define sin ↾[0, 2π]. In his thesis [5], Günaydın shows that for any nonzero α ∈ R, both (Re, e) and (Rs, e) define Q and hence also Z. We now extend his result. Corollary 3. If G is noncyclic and C is an infinite convex subset of e, then G is definable in (Re, C). If moreover G has finite rank, then Z is definable. Proof. There exist 0 < a < b < ∞ such that [a, b] ∩ C is infinite. As log ↾[a, b] is definable in Re, we have [log a, log b]∩G definable in (Re, C). Apply Propositions 1 and 2. □ Corollary 4. If e is infinite and A is a nondegenerate arc of the unit circle, then G + 2πZ is definable in (Rs, e iG ∩ A). If moreover G has finite rank, then Z is definable. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use DEFINING SUBGROUPS 3475 Proof. Intersect eiG∩A with either the right half-plane or the left half-plane, project onto the second coordinate, and then take the image under arcsin. This set is definable and contains an infinite convex subset of G+ 2πZ. Apply Proposition 1. If G has finite rank, then so does G+ 2πZ. Apply Proposition 2. □ We have concentrated so far on expanding by single groups, but there are many reasons to be interested in expanding by collections of groups. The most fundamental question is this: For Q-linearly independent α, β ∈ R, what can be said about (R, e, e)? Other than the obvious fact that the direct product e is definable, very little known is known as of this writing. The situation is similar for the structures (R, e, e) and (R, e, e). We now provide some partial answers. Corollary 5. If α, β ∈ R are Q-linearly independent, then Z is definable in each of the structures (Re, e , e), (Rs, e , e), (Re, e , e), (Rs, e , e). Proof. Since the product groups are definable, the first two cases are immediate from Corollaries 3 and 4. For the third, observe that e ∩R>0 and e ∩ R are definable, thus reducing to the first case. For the fourth, project e(1+iα)R∩ e onto the unit circle and apply Corollary 4. □ As an immediate consequence, we have the following: Corollary 6. Let r ∈ R ∖Q. Then Z is definable in each of the structures (Re, x , e), (Re, x , e), (Rs, x , e). We close with one more application. In [8], Miller and Speissegger establish a trichotomy for expansions of the structure Ran by individual trajectories of certain kinds of analytic planar vector fields. By combining Corollaries 5 and 6 with Corollary 4 of their paper, the trichotomy extends to expansions of Ran by arbitrary collections of such trajectories.