Chris Miller

Geometric categories and o-minimal structures

The theory of subanalytic sets is an excellent tool in various analytic-geometric contexts; see, for example, Bierstone and Milman [1]. Regrettably, certain “nice” sets—like { (x, x) : x > 0 } for positive irrational r, and { (x, e−1/x) : x > 0 }—are not subanalytic (at the origin) in R. Here we make available an extension of the category of subanalytic sets that has these sets among its objects, and that behaves much like the category of subanalytic sets. The possibility of doing this emerged in 1991 when Wilkie [27] proved that the real exponential field is “model complete”, followed soon by work of Ressayre, Macintyre, Marker and the authors; see [21], [5], [7] and [19]. However, there are two obstructions to the use by geometers of this development: (i) while the proofs in these articles make essential use of model theory, many results are also stated there (efficiently, but unnecessarily) in model-theoretic terms; (ii) the results of these papers apply directly only to the cartesian spaces R, and not to arbitrary real analytic manifolds. Consequently, in order to carry out our goal we recast here some results in those papers—as well as many of their consequences—in more familiar terms, with emphasis on results of a geometric nature, and allowing arbitrary (real analytic) manifolds as ambient spaces. We thank W. Schmid and K. Vilonen for their suggestion that this would be a useful undertaking; indeed, they gave us a “wish list” (inspired by Chapters 8 and 9 of Kashiwara and Schapira [12]; see also §10 of [22]) which strongly influenced the form and content of this paper. We axiomatize in Section 1 the notion of “behaving like the category of subanalytic sets” by introducing the notion of “analytic-geometric category”. (The category Can of subanalytic sets is the “smallest” analytic-geometric category.) We also state in Section 1 a number of properties shared by all analytic-geometric categories. Proofs of the more difficult results of this nature, like the Whitney-stratifiability of sets and maps in such a category, often involve the use of charts to reduce to the case of subsets of R. For subsets of R, there already exists the theory of “o-minimal structures on the real field” (defined in Section 2); this subject is developed in detail in [4] and is an abstraction of

Expansions of dense linear orders with the intermediate value property

Let 9l be an expansion of a dense linear order (R, <) without endpoints having the intermediate value property, that is, for all a, b c R, every continuous (parametrically) definable function f: [a, b] -* R takes on all values in R between f (a) and f (b). Every expansion of the real line (R. <), as well as every o-minimal expansion of (R, <), has the intermediate value property. Conversely, some nice properties, often associated with expansions of (R, <) or with o-minimal structures, hold for sets and functions definable in N. For example, images of closed bounded definable sets under continuous definable maps are closed and bounded (Proposition 1.10). Of particular interest is the case that 9l expands an ordered group, that is, X defines a binary operation * such that (R, <, *) is an ordered group. Then (R, *) is abelian and divisible (Proposition 2.2). Continuous nontrivial definable endomorphisms of (R, *) are surjective and strictly monotone, and monotone nontrivial definable endomorphisms of (R, *) are strictly monotone, continuous and surjective (Proposition 2.4). There is a generalization of the familiar result that every proper noncyclic subgroup of (R, +) is dense and codense in R: If G is a proper nontrivial subgroup of (R, *) definable in 9A, then either G is dense and codense in R, or G contains an element u such that (R, <, *, e, u, G) is elementarily equivalent to (Q, <, +, 0, 1, Z), where e denotes the identity element of (R, *) (Theorem 2.3). Here is an outline of this paper. First, we deal with some basic topological results. We then assume that X expands an ordered group and establish the results mentioned in the preceding paragraph. Some examples are then given, followed by a brief discussion of analytic results and possible limitations. In an appendix, an explicit axiomatization (used in the proof of Theorem 2.3) is given for the complete theory of the structure ((Q, <, +, 0, , 1). Conventions. From now on, we assume that 9l has the intermediate value property (IVP for short). Formally adjoin endpoints -0o and +oo to R in the usual fashion and put R,, = R U {?oo}. AsetA C Risconvexifforalla,b c Awitha < b, theset{x c R: a < x < b} is contained in A. An interval (in R) is a convex set I C R such that both inf I and sup I exist in R,, (that is, I has endpoints in RJ). The usual notation is employed for the various kinds of intervals. All Cartesian powers Rn are equipped with the

Structures having o-minimal open core

The open core of an expansion of a dense linear order is its reduct, in the sense of definability, generated by the collection of all of its open definable sets. In this paper, expansions of dense linear orders that have o-minimal open core are investigated, with emphasis on expansions of densely ordered groups. The first main result establishes conditions under which an expansion of a densely ordered group has an o-minimal open core. Specifically, the following is proved: Let R be an expansion of a densely ordered group (R, <, *) that is definably complete and satisfies the uniform finiteness property. Then the open core of R is o-minimal. Two examples of classes of structures that are not o-minimal yet have o-minimal open core are discussed: dense pairs of o-minimal expansions of ordered groups, and expansions of o-minimal structures by generic predicates. In particular, such structures have open core interdefinable with the original o-minimal structure. These examples are differentiated by the existence of definable unary functions whose graphs are dense in the plane, a phenomenon that can occur in dense pairs but not in expansions by generic predicates. The property of having no dense graphs is examined and related to uniform finiteness, definable completeness, and having o-minimal open core.

Borel subrings of the reals

A Borel (or even analytic) subring of R either has Hausdorff dimension 0 or is all of R. Extensions of the method of proof yield (among other things) that any analytic subring of C having positive Hausdorff dimension is equal to either R or C.

A growth dichotomy for o-minimal expansions of ordered groups

Let R be an o-minimal expansion of a divisible ordered abelian group (R, <, +, 0, 1) with a distinguished positive element 1. Then the following dichotomy holds: Either there is a 0-definable binary operation · such that (R, <, +, ·, 0, 1) is an ordered real closed field; or, for every definable function f : R → R there exists a 0-definable λ ∈ {0} ∪ Aut(R, +) with limx→+∞[f(x) − λ(x)] ∈ R. This has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure M := (M, <, . . . ) there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order) M-definable groups with underlying set M . R. Poston showed in [8] that given an o-minimal expansion R of (R, <, +), if multiplication is not definable in R, then for every definable function f : R → R there exist r, c ∈ R such that limx→+∞[f(x) − rx] = c. In this paper, this fact is generalized appropriately for o-minimal expansions of arbitrary ordered groups. We say that an expansion (G, <, ∗, . . . ) of an ordered group (G, <, ∗) is linearly bounded (with respect to ∗) if for each definable function f : G → G there exists a definable λ ∈ End(G, ∗) such that ultimately |f(x)| ≤ λ(x). (Here and throughout, ultimately abbreviates “for all sufficiently large positive arguments”.) We now list the main results of this paper. Let R := (R, <, . . . ) be o-minimal. Theorem A (Growth Dichotomy). Suppose that R is an expansion of an ordered group (R, <, +). Then exactly one of the following holds: (a) R is linearly bounded; (b) R defines a binary operation · such that (R, <, +, ·) is an ordered real closed field. If R is linearly bounded, then for every definable f : R → R there exist c ∈ R and a definable λ ∈ {0} ∪ Aut(R, +) with limx→+∞[f(x)− λ(x)] = c. Theorem B. Suppose that R is a linearly bounded expansion of an ordered group (R, <, +, 0, 1) with 1 > 0. Then every definable endomorphism of (R, +) is 0-definable. If R′ (with underlying set R′) is elementarily equivalent to R, then the ordered division ring of all R′-definable endomorphisms of (R′, +) is canonically isomorphic to the ordered division ring of all R-definable endomorphisms of (R, +). The growth dichotomy imposes some surprising constraints on continuous definable groups with underlying set R. (Here and throughout, all topological notions are taken with respect to the product topologies induced by the order topology.) Received by the editors June 5, 1996. 1991 Mathematics Subject Classification. Primary 03C99; Secondary 06F20, 12J15, 12L12. The first author was supported by NSF Postdoctoral Fellowship No. DMS-9407549. c ©1998 American Mathematical Society

Expansions of o-minimal structures by dense independent sets

Abstract Let M be an o-minimal expansion of a densely ordered group and H be a pairwise disjoint collection of dense subsets of M such that ⋃ H is definably independent in M . We study the structure ( M , ( H ) H ∈ H ) . Positive results include that every open set definable in ( M , ( H ) H ∈ H ) is definable in M , the structure induced in ( M , ( H ) H ∈ H ) on any H 0 ∈ H is as simple as possible (in a sense that is made precise), and the theory of ( M , ( H ) H ∈ H ) eliminates imaginaries and is strongly dependent and axiomatized over the theory of M in the most obvious way. Negative results include that ( M , ( H ) H ∈ H ) does not have definable Skolem functions and is neither atomic nor satisfies the exchange property. We also characterize (model-theoretic) algebraic closure and thorn forking in such structures. Throughout, we compare and contrast our results with the theory of dense pairs of o-minimal structures.

Avoiding the projective hierarchy in expansions of the real field by sequences

Some necessary conditions are given on infinitely oscillating real functions and infinite discrete sets of real numbers so that first-order expansions of the field of real numbers by such functions or sets do not define N. In particular, let f: R → R be such that lim x→+∞ f(x) = + ∞ , f(x) = O(e xN ) as x → +∞ for some N ∈ N, (R,+,·,f) is o-minimal, and the expansion of (R, +,·) by the set { f(k): k ∈ N } does not define N. Then there exist r > 0 and P ∈ R[x] such that f(x) = e P(x) (1 + O(e -rx )) as x → +∞.

Accountability Policy Implementation and the Case of Smaller School District Capacity: Three Contrasting Cases that Examine the Flow and Use of NCLB Accountability Data

The No Child Left Behind Act increases pressure on schools and districts to use standardized state test data. Seeking to learn about the process of turning accountability data into actionable information, this paper presents findings from three case studies of small to medium sized school districts. The study examines the flow of state science test data through the district central offices to schools and teachers, revealing how the data is used and how districts are adapting to increased pressures for data use. The findings show the importance of district central office capacity and the degree of centralization of curriculum management.

Basics of O-minimality and Hardy Fields

This paper consists of lecture notes on some fundamental results about the asymptotic analysis of unary functions definable in o-minimal expansions of the field of real numbers.

A dichotomy for expansions of the real field

A dichotomy for expansions of the real field is established: Either the set of integers is definable or every nonempty bounded nowhere dense definable subset of the real numbers has Minkowski dimension zero.