View of ﬁrst-Order Logic and O-Minimality

TAMENESS RESULTS FOR EXPANSIONS OF THE REAL FIELD BY GROUPS

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of

Philosophy in the Graduate School of the Ohio State University

Michael A. Tychonievich, B.S., M.S.

Graduate Program in Mathematics

The Ohio State University

2013

Dissertation Committee:

Prof. Christopher Miller, Advisor

Prof. Timothy Carlson

Prof. Ovidiu Costin

Prof. Daniel Verdier c Copyright by

Michael A. Tychonievich

2013 ABSTRACT

Expanding on the ideas of o-minimality, we study three kinds of expansions of the real ﬁeld and discuss certain tameness properties that they possess or lack. In

Chapter 1, we introduce some basic logical concepts and theorems of o-minimality.

In Chapter 2, we prove that the ring of integers is definable in the expansion of the real field by an infinite convex subset of a finite-rank additive subgroup of the reals.

We give a few applications of this result. The main theorem of Chapter 3 is a structure theorem for expansions of the real field by families of restricted complex power functions. We apply it to classify expansions of the real field by families of locally closed trajectories of linear vector fields. Chapter 4 deals with polynomially bounded o-minimal structures over the real field expanded by multiplicative subgroups of the reals. The main result is that any nonempty, bounded, definable d-dimensional submanifold has finite d-dimensional Hausdorff measure if and only if the dimension of its frontier is less than d.

ii To my father, who has fallen asleep.

iii ACKNOWLEDGMENTS

As a graduate student, I owe thanks to many people, but I will only list a few here. I thank my adviser Prof. Chris Miller for his guidance and feedback. I thank my committee members Prof. Tim Carlson and Prof. Ovidiu Costin for their eﬀorts as well. I also thank Prof. Patrick Speissegger and Prof. Lou van den Dries for their valued discussions, and the organizers of the Fields Institute Thematic Program on

O-minimal Structures and Real Analytic Geometry (January–June 2009) for orga- nizing such a great event. I thank Prof. Philipp Hieronymi for his friendship and collaboration. I am thankful for the support staﬀ of the Ohio State Mathematics De- partment, especially Cindy Bernlohr, John Lewis, and Denise Witcher; I thank the whole department for supporting me while I pursued this work, especially Prof. Janet

Best. Finally, a big thank you goes out to Prof. Thomas Lemberger and Dr. Aaron

Pesetski for introducing me to the realities of academic research as an undergraduate.

I thank my father for encouraging me to pursue mathematics and science at an early age, and my mother for instilling in me a sense of responsibility. The rest of my family helped support me throughout my education, especially my brother Dr. John

Tychonievich and my niece and nephew, Abbey and Eddie. I thank my friends, especially Josh and Amanda, for their support and companionship, and Kirsten, for helping me to align my goals with my values.

iv VITA

2004 ...... B.S., The Ohio State University

2007 ...... M.S. in Mathematics, The Ohio State University

2004-Present ...... Graduate Teaching Associate, The Ohio State University

PUBLICATIONS

Tychonievich, Michael, Deﬁning additive subgroups of the reals from convex subsets, Proceedings of the American Mathematical Society 137 (2009), 3473–3476.

Tychonievich, Michael, The set of restricted complex exponents for expansions of the reals, Notre Dame Journal of Formal Logic 53 (2) (2012), 175–186.

Hieronymi, Philipp; Tychonievich, Michael, Interpreting the projective hierarchy in expansions of the real line, to appear in Proceedings of the American Mathematical Society, preprint available at http://arxiv.org/abs/1203.6299.

v FIELDS OF STUDY

Major Field: Mathematics

Specialization: Model Theory

vi TABLE OF CONTENTS

Abstract ...... ii

Dedication ...... iii

Acknowledgments ...... iv

Vita...... v

CHAPTER PAGE

1 Introduction ...... 1

1.1 Basics of ﬁrst-order logic ...... 1 1.2 Syntactic conditions ...... 5 1.3 O-minimality ...... 8

2 Deﬁning Additive Subgroups of the Reals from Convex Subsets . . . . . 11

2.1 Introduction and main result ...... 11 2.2 Applications ...... 13

3 The Set of Restricted Complex Exponents for Expansions of the Reals . 16

3.1 Introduction and preliminaries ...... 16 3.2 Proof of main result ...... 18 3.3 Linear vector ﬁelds ...... 24 3.4 Complex powers on subsets of C ...... 26 3.5 Invariance of E under elementary equivalence ...... 29 3.6 Optimality ...... 31

4 Metric Properties of Sets Deﬁnable in (R, α−N) ...... 34

4.1 Introduction ...... 34 4.2 Induced structure ...... 41 4.3 Deﬁnable estimates for d-volume ...... 47 4.4 Dimension and volume ...... 51

vii Bibliography ...... 53

viii CHAPTER 1

INTRODUCTION

We begin with a brief review of ﬁrst-order logic and o-minimality. We follow loosely the path of the ﬁrst few chapters of Marker [18] for basic logic. An introduction to much of the same material from a more mathematical perspective can be found in

Miller [24], which we will not use here because certain syntactical conditions we will need further on are diﬃcult to express using the method of Miller’s exposition in that paper. We will follow with a review of some important results in o-minimality that are contained in van den Dries [9].

1.1 Basics of ﬁrst-order logic

A (ﬁrst-order) language L consists of three sets: a set of function symbols F along with positive integers nf for each f ∈ F, a set of relation symbols R along with positive integers nR for each R ∈ R, and a set of constant symbols C. Here a symbol is just an element of a set, all symbols are pairwise distinct, and no symbol belongs to

{(, ), ¬, ∨, ∧, ∃, ∀}, the set of logical operators. The integers nf and nR are referred to as the arities of their respective function or relation symbols. For a fixed language L, an L-structure M is a set M, along with an interpretation for each of the symbols of the language L. An interpretation of a function symbol f is a specified function f M : M nf → M, an interpretation of a relation symbol R is a specified subset

1 RM ⊆ M nR , and an interpretation of a constant symbol c is an element cM ∈ M.

The set M is called the underlying set of the structure M. An interpretation can be considered a function from the set of symbols of the language L to the system of sets

n {P(M ): n ∈ N}, where P is the power set operator, by identifying each function with its graph.

A language will usually be given by listing all of its symbols, while a structure will be given by listing the underlying set of the structure along with the interpretation of each symbol in its language. For example, the language L of rings is {+, −, ·, 0, e}, and the ring of integers is the L-structure (Z, +, −, ·, 0, 1), indicating that Z is the underlying set of this structure, that +, −, and · are interpreted as integer addition, subtraction, and multiplication respectively in this structure, and that 0 and e is interpreted as the numbers 0 and 1 respectively. A language L0 expands a language

L if every symbol of L is a symbol of the same type and arty in L0. In this case, an L0-structure M0 is an expansion of a L-structure M if both structures have the same underlying set, and every symbol in L has the same interpretation in both structures.

Let L be a language and let {vn}n∈N be distinct variable symbols that are not symbols for the language L. The set of L-terms is the smallest set T that contains each variable symbol and each constant symbol, and such that for each function

symbol f of the language L and each sequence of L-terms t1, . . . , tnf , the string of

symbols f(t1, . . . , tnf ) is in T . We extend our interpretation to T inductively by setting (f(t , . . . , t ))M = f M(tM, . . . , tM). 1 nf 1 nf An atomic L-formula is any string of symbols either of the form s = t, where s

and t are L-terms, or of the form R(t1, . . . , tnR ), where R is an nR-ary relation symbol in L. The set of L-formulas is the closure of the set of atomic L-formulas under the following operations:

2 • If φ and ψ are L-formulas, then so are (¬φ), (φ ∧ ψ), and (φ ∨ ψ).

• If φ is an L-formula, then for all i ∈ N so are (∃viφ) and (∀viφ).

For formulas ψ and ψ, we abbreviate (¬φ) ∨ ψ as ψ → φ and (ψ → φ) ∧ (φ → ψ) as

ψ ↔ φ.

An occurrence of a variable symbol v is considered bound in a formula if it occurs within the parentheses immediately following the quantiﬁer symbols ∃v or ∀v, and an occurrence of a variable symbol is said to be free if it is not a bound occurrence. We require that in each formula where a variable symbol vi occurs, the variable is either free in every occurrence or that every occurrence is bound by the same quantiﬁer symbol.

We will now deﬁne what it means for a structure to be a model, so that we can, for instance, state when a structure in the language of rings is in fact a ring. For

any ψ that is either a L-formula with free variables from the list v = (vi1 , . . . , vin )

or an L-term involving only variables from the list v = (vi1 , . . . , vin ), and any a =

n (a1, . . . , an) ∈ M , we write ψ(a) to indicate that for each k ∈ {1, . . . n}, each free

occurrence of vik in ψ is replaced by ak (more precisely, new symbols for a1, . . . , an must be added to the language ﬁrst).

Let M be an L-structure, let φ be an L-formula with free variables from the

n list v = (vi1 , . . . , vin ), and let a = (a1, . . . , an) ∈ M . We now deﬁne the statement M |= φ(a), which is read as “M satisﬁes φ(a)”, “φ(a) is true in M”, or “M models

φ(a)”.

• If φ is s = t for L-terms s and t, then M |= φ(a) if and only if sM(a) = tM(a).

• If φ is R(t1, . . . , tnR ) for a relation symbol R and L-terms t1, . . . , tnR , then M |= φ(a) if and only if (tM(a), . . . , tM (a)) ∈ RM. 1 nR

• If φ is (¬ψ) for some L-formula ψ, then M |= φ(a) if and only if M 6|= ψ(a). 3 • If φ is (ψ ∧ θ) for some L-formulas ψ and θ, then M |= φ(a) if and only if

M |= ψ(a) and M |= θ(a).

• If φ is (ψ ∨ θ) for some L-formulas ψ and θ, then M |= φ(a) if and only if

M |= ψ(a) or M |= θ(a).

• If φ is (∃vmψ) for some L-formula ψ, m∈ / {i1, . . . , in}, and ψ only involves free

variables from the list (v, vm), then M |= φ(a) if and only if there is a b ∈ M such that M |= ψ(a, b).

• If φ is (∀vmψ) for some L-formula ψ, m∈ / {i1, . . . , in}, and ψ only involves free

variables from the list (v, vm), then M |= φ(a) if and only if M |= ψ(a, b) for every b ∈ M.

To increase readability, we will leave oﬀ excess parentheses whenever this does not result in ambiguity.

An L-sentence is deﬁned to be an L-formula with no free variables, and an L- theory is deﬁned to be a set of L-sentences. For a theory T , we write M |= T to indicate that M |= φ for every φ ∈ T . In this situation, we say that M is a model of T . For two theories S and T , we write S |= T to indicate that any model of S is a model of T , and we say that T is a set of consequences of S. If both S |= T and

T |= S, then we say that S axiomatizes the theory T . For a given L-structure M, the theory Th(M) is deﬁned to be the set of all L-sentences that are true in M, and we say that a theory axiomatizes the structure M if it axiomatizes Th(M).

Let X ⊆ M n. We say that X is deﬁnable (in M) if and only if there is an

L-formula φ with free variables from the list v = (v1, . . . , vn, vn+1, . . . , vn+m) and an element b ∈ M m such that X = {a ∈ M n : M |= φ(a, b)}. In this case, we say that

φ(v, b) defines X. If more precision is needed, we will say that X is definable with parameters from A if the element b in the above definition can be taken to be an 4 element of Am, and we say that X is ∅-definable if m may be taken to be 0 in the above definition (or, equivalently, if we may take A = ∅). A function is definable with parameters from A if and only if its graph is definable with parameters from A.

The notion of definability yields an equivalence relation between structures in the following way. Suppose that L and L0 are languages and that M and M0 are structures with the same underlying set. We say that M and M0 are interdefinable if and only exactly the same sets are definable in each structure. Interdefinability is a useful notion in later chapters, as it will allow us to alter the language of a structure to one that allows stricter syntactic conditions, such as those described in the next section.

1.2 Syntactic conditions

Now let us look at a few conditions that we may impose on languages, structures, and theories. Unless stated otherwise, we assume that all structures share a common language and that all language-dependent deﬁnitions refer to that language.

We say that a formula is quantiﬁer free if it contains no instances of ∃ or ∀.

Suppose that T is a theory whose language contains a constant symbol. The theory T is said to admit quantifier elimination if and only if for every formula φ(v) in the language of T such that the free variables of φ(v) are among the tuple of variables v, there is a quantifier-free formula ψ(v)in the language of T such that the free variables of φ(v) are among the tuple of variables v and such that T |= ∀v(φ(v) ↔ ψ(v)), where ∀v denotes universal quantification over each of the variables in the tuple v, in order. A structure is said to admit quantifier elimination if its theory admits quantifier elimination. For a given structure, there is always a structure that is definably equivalent to it that admits quantifier elimination: expand the language of the structure by a symbol for each definable set.

5 From now on, we let R denote the ordered ﬁeld of real numbers in the language {<, +, −, ·, 0, 1}. Tarski [31] showed that the theory of this structure admits quantiﬁer

Z elimination. Van den Dries [6] proved that the structure (R, {2 }) admits quantiﬁer elimination in the language {<, +, ·, 0, 1, {Gn}n∈Z+ }, where Gn is interpreted as the subgroup 2nZ of 2Z.

If a structure admits quantifier elimination in a language with no relation symbols, then definable sets are exactly those sets that are finite unions of solution sets to finite systems of equations and inequations among functions that correspond to terms. This viewpoint helps us to understand the geometric meaning of quantifier elimination: as the theory of algebraically closed fields admits quantifier elimination in the language of fields (see [18, 3.2]), one sees that any definable set in an algebraically closed field is a boolean combination of (not necessarily irreducible) affine algebraic varieties.

Weaker than quantiﬁer elimination is the related notion of model completeness.

We say that an L-formula is existential if it is of the form ∀v1∀v2 ... ∀vnφ where φ is a quantiﬁer-free L-formula Again suppose that T is a theory whose language contains a constant symbol. The theory T is said to be model complete if and only if for every formula φ(v) in the language of T such that the free variables of φ(v) are among the tuple of variables v, there is an existential formula ψ(v) in the language of

T such that the free variables of φ(v) are among the tuple of variables v and such that

T |= ∀v(φ(v) ↔ ψ(v)). A structure is said to be model complete if its theory is model complete. Since every quantifier-free formula is existential, a theory that admits quantifier elimination is automatically model complete, but the converse may not be true; the theory of dense linear orders with endpoints in the language {<} is model complete, but does not admit quantifier elimination. If there are no relation symbols in a language, model completeness can be interpreted in the following way: any definable set is expressible as a finite union of projections of solution sets of systems

6 of equations and inequations between functions that correspond to terms. Another way of saying this is that every deﬁnable set is a ﬁnite union of basic sets, each of which is the set of parameters for which a system of equations and inequations among functions corresponding to terms has a solution. In this way, model completeness corresponds to the Implicit Function Theorem.

We say that an L-formula is universal if it is of the form ∃v1∃v2 ... ∃vnφ where φ is a quantifier-free L-formula, and that a theory is universal if all of its sentences are universal formulas. A theory is universally axiomatizable if it is axiomatized by a universal theory. The concept of universal axiomatization is related to that of Skolem functions. For a theory T and a formula φ with free variables from among those in the (n + 1)-tuple of variables (x1, . . . , xn, v), a definable Skolem function for φ (with respect to T ) is a formula ψ(x, v) that defines the graph of a function in every model of T , and such that T |= ∃vφ(x, v) if and only if T |= ∃v(φ(x, v) ∧ ψ(x, v)). If every formula has a definable Skolem function with respect to T , then we say that T has definable Skolem functions. We say that a structure has definable Skolem functions if its theory does. Given an L-structure M with definable Skolem functions, let L0 be the language obtained by extending L by a function symbol for each definable Skolem function. Let M0 be the L0-structure arrived at by interpreting these function symbols as the appropriate Skolem functions. Then M0 and M are clearly interdefinable. As any existential sentence is witnessed uniformly by an appropriate term, the theory

Th(M0) is universally axiomatizable. Intuitively, this is how one may think about universal axiomatizability: existential claims are witnessed by terms.

When universal axiomatization and model completeness coincide in a structure, we have the following important result, apparently due to Herbrand [15]:

Theorem 1.2.1. Let L be a language and let M be an L-structure that is universally axiomatizable and model complete. Then M admits quantiﬁer elimination and for any

7 n deﬁnable function f : M → M there are terms t1, . . . , tm with free variables from among (v1, . . . , vn) such that

m ! _ M |= ∀x f(x) = tk(x) . k=1

That is, f is given piecewise by the terms t1, . . . , tm.

For the rest of this dissertation, we assume that the reader is conversant in the basic notions of model theory, even those not stated explicitly in this Chapter. Again, one may consult Marker [18] for this information.

1.3 O-minimality

Now let L be a language with the binary relation symbol < and let M be a L-structure. Following Pillay and Steinhorn’s foundational work [27] inspired by of the ideas of van den Dries [5], we say that M is an o-minimal structure if and only if (M, <) is a linear order on M and every definable subset of M is a finite union of points and open intervals. Here an open interval is a set of the form (a, b) = {x ∈ M : a < x < b} for some a, b ∈ M. If M is o-minimal, the only definable subsets of M are those that would be definable if L only contained the relation symbol < that is interpreted as a linear order in M. It is in this sense that the structure is minimal with respect to order; the “o” in o-minimal stands for order. A mathematical introduction to the subject of o-minimality is van den Dries’s book [9]. We will only review a few highlights here.

Cell Decomposition

A fundamental result in o-minimality is the cell decomposition theorem, which deals with the structure of definable subsets of Cartesian powers M n. First, we inductively define what cells are. A subset of M is a 0-cell if it is a point, and a subset of M is 8 a 1-cell if it is a nonempty open interval. Suppose that n ∈ N is such that we have defined what a k-cell in M n is. Then for any definable set Y ⊆ M n+1 we say that

• Y is a k-cell if there is a k-cell X ⊆ M n and a continuous deﬁnable function

f : X → M such that Y is the graph of f,

• Y is a (k+1)-cell if there is a k-cell X ⊆ M n and a continuous deﬁnable function

f : X → M such that Y = {(x, y) ∈ X × M : f(x) < y} or Y = {(x, y) ∈

X × M : f(x) > y},

• Y is a (k+1)-cell if there is a k-cell X ⊆ M n and continuous deﬁnable functions

f, g : X → M such that f < g and Y = {(x, y) ∈ X × M : f(x) < y < g(x)},

• Y is a (k + 1)-cell if there is a k-cell X ⊆ M n such that Y = X × M.

Theorem 1.3.1 (Pillay and Steinhorn [27]). Let M be an o-minimal structure. Then every deﬁnable subset of M n is a ﬁnite union of cells.

Later researchers have expanded on this result in various directions to produce much stronger cell-decomposition results, some of which we will discuss in Chapter 4.

Further analysis of the cell decomposition theorem gives us much more information

n about how complicated deﬁnable sets in an o-minimal structure may be. Let X ⊆ R be a nonempty deﬁnable and let x ∈ X. We say that X has dimension d at x if for any

n open ball B around x there is a projection map from R to a d-dimensional plane such that the projection of X ∩ B has nonempty interior, but for any open ball B around

n+1 X and any projection from R to any (d + 1)-dimensional plane, the projection of X ∩ B has empty interior. We say that the dimension of a nonempty deﬁnable set X is the number dim(X) = sup{d : X has dimension d at x for some x ∈ X}. Finally, we set dim(∅) = −∞. It is immediate from cell decomposition that for a nonempty

9 deﬁnable set X, dim(X) is the largest integer k such that X contains a k-cell as a subset.

We write cl(X) to denote the topological closure of X, and we deﬁne the frontier of X as fr(X) = cl(X) \ X. An important consequence of the cell decomposition theorem is the following:

Theorem 1.3.2. Let X be a nonempty set deﬁnable in an o-minimal structure M and let d = dim X. Then d > dim(fr X). If X is bounded and M is an expansion of

(R, <), then the d-dimensional Hausdorﬀ measure of X is ﬁnite.

The main result of Chapter 4 is Theorem 4.1.1, a version of this result for certain non-o-minimal expansions of the real ﬁeld.

10 CHAPTER 2

DEFINING ADDITIVE SUBGROUPS OF THE REALS

FROM CONVEX SUBSETS

2.1 Introduction and main result

In this chapter1 we answer some questions about expansions of o-minimal structures on the real field R by various groups. For more detailed treatment of some of the issues raised below, see van den Dries and Günaydın [11] and Miller [21, 22]. Throughout this chapter, let K = (K, +, ·, < ) be an ordered subfield of R, and let G be a nontrivial additive subgroup of K.

Theorem 2.1.1. Let C ⊆ G be inﬁnite and convex in G. Then G is deﬁnable in the structure (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 sC ∩ [0, ε) ⊆ C. By density of G, there exists c ∈ C ∩ (0, ε). Note that

1Some version of these results has previously appeared in Tychonievich [32]. 11 sc ∈ C ⊆ G. Let k be the least 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.

Theorem 2.1.2. If G has finite rank (equivalently, if the Q-linear span of G is finite dimensional), and if C ⊆ G is infinite and convex, then (K,C) defines Z.

Proof. By Theorem 2.1.1 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 [28], Z is definable in (J, +, ·). Since J is definable in (K,R), so is Z.

Remarks. (i) If G is finitely generated and 1 ∈ G, then (K,G) defines Z for all subrings K of R containing G; see Miller [21, 6.1]. (ii) The set R defined in the proof of Proposition 2.1.2 contains the set S as in the proof of Proposition 2.1.1; equality holds if and only if G is dense.

(iii) Any conditions on G that force the ﬁeld J to be either a ﬁnite-degree algebraic

0 extension of Q or a purely transcendental extension of some J ⊆ J yield definability of Z, again by [28] for the former and by R. Robinson [29] for the latter. (iv) 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 [10]. We do not know of any G other than real-closed subfields that do not define Z over R.

(v) Of course, undecidability of Th(K,C) follows from the definability of Z, but much more is true when K = R. The structure (R, Z) defines every set in the so-called real projective hierarchy. In this structure, every open set is definable, as is every Borel set, every projection of a Borel set, etc. Problems involving the structure (R, Z) are

12 dealt with in the subject of descriptive set theory; see Kechris [17] for more details and a review of Borel coding.

2.2 Applications

We now concentrate on the case that K = R and consider expansions of R by multi- ∼ 2 plicative subgroups of C\{0}. We make the usual identiﬁcations R ,→ C = R . Given Z Z ⊆ C, let e denote the image of Z under complex exponentiation. We focus here wG on expansions of structures of the form (R, (e )w∈W ) where W ⊆ C. At present, there are essentially only two positive results known:

• van den Dries and G¨unaydın [11]. If G is noncyclic and has ﬁnite rank, then

G iG neither (R, e ) nor (R, e ) deﬁne Z. Indeed, every deﬁnable set (of any arity)

is a boolean combination of Fσ sets.

• Miller [21,22]. If G is cyclic and R is an o-minimal expansion of R that defines G 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 >0 R deﬁnes the restriction of the function t 7→ sin(ω log t): R → R to some >0 G (1+iω) nontrivial subinterval of R . Then (R, e ) deﬁnes the group e R. By van

den Dries [7], the expansion of R by the restriction sin(ω log t)[1, 2] is o-minimal i and deﬁnes no irrational power functions. Thus (R, e R) does not deﬁne Z.

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 van den Dries [7], both Re and Rs are o-minimal and define no irrational power functions. Indeed, the 13 same is true of their amalgam (R, exp [0, 1], sin [0, 1]). It is worth noting that, by

Bianconi [2], Rs does not deﬁne exp [0, 1], and Re does not deﬁne sin [0, 2π]. In his

αQ iαQ thesis, Günaydın [14] shows that for any nonzero α ∈ R, both (Re, e ) and (Rs, e ) define Q, hence also Z. We now extend Günaydın’s result.

Corollary 2.2.1. If G is noncyclic and C is an infinite convex subset of eG, 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 Theorems 2.1.1 and 2.1.2.

Corollary 2.2.2. If eiG is inﬁnite and A is a nondegenerate arc of the unit circle,

iG then G + 2πZ is definable in (Rs, e ∩ A). If moreover G has finite rank, then Z is definable.

Proof. Intersect eiG ∩ A with either the right half-plane or the left half-plane, project on the second coordinate, and then take the image under arcsin. This set is deﬁnable and contains an inﬁnite convex subset of G + 2πZ. Apply Theorem 2.1.1. If G has

ﬁnite rank, then so does G + 2πZ. Apply Theorem 2.1.2.

We have concentrated so far on expanding by single groups, but there are many reasons to be interested in expanding by collections of groups.

Corollary 2.2.3. If α, β ∈ R are Q-linearly independent, then Z is deﬁnable in each of the structures

αZ βZ iαZ iβZ (1+iα)R (1+iβ)R (1+iα)R (1+iβ)R (Re, e , e ), (Rs, e , e ), (Re, e , e ), (Rs, e , e ).

Proof. Since the various product groups are deﬁnable, the ﬁrst two cases are imme-

(1+iα) >0 diate from Corollaries 2.2.1 and 2.2.2. For the third, observe that e R ∩ R and 14 (1+iβ) >0 e R ∩ R are deﬁnable, thus reducing to the ﬁrst case. For the fourth, project e(1+iα)R ∩ e(1+iβ)R onto the unit circle and apply Corollary 2.2.2.

As an immediate consequence,

Corollary 2.2.4. Let r ∈ R \ Q. Then Z is deﬁnable in each of the structures

r αZ r (1+iα)R r (1+iα)R (Re, x , e ), (Re, x , e ), (Rs, x , e ).

Note. After the results in this chapter were ﬁrst published [32], the applications in

Corollaries 2.2.3 and 2.2.4 were improved upon by Hieronymi [16]; all of the conclusions of those corollaries hold with Re and Re replaced by R. See Proposition 3.2.4 in the next chapter for details. Corollary 2.2.1 has been directly applied by Miller [23] in his classification results for structures that expand the reals by trajectories of vector fields. Theorem 3.3.1 in the next chapter of this dissertation is part of this classification program.

15 CHAPTER 3

THE SET OF RESTRICTED COMPLEX EXPONENTS

FOR EXPANSIONS OF THE REALS

3.1 Introduction and preliminaries

In this chapter1 we extend the notion of the ﬁeld of exponents for expansions of the

ﬁeld of real numbers, introduced by Miller [19], to allow for complex exponents. In

RE this we seek to better understand the deﬁnable functions of the structure R :=

(R, exp [0, 1], sin [0, 1]), in particular the model-theoretic implications of expanding o-minimal (and other tame) structures on the real ﬁeld by such functions. Here “RE” stands for “restricted elementary;” we direct the reader to van den Dries [8] for more

RE information about R and again to van den Dries [9] for basic results in o-minimality. Throughout this chapter, “definable” means first-order definable with parameters and a “term” is a term in the language of the structure being considered. We write

R for the ordered field of real numbers (R, +, −, ·, <, 0, 1), and we write C to mean the field of complex numbers as a definable object in R. a+ib a For x > 0 and a, b ∈ R, we put x = x (cos b log x + i sin b log x), where log denotes the real logarithm. For an expansion R of R, we say that a+ib is a restricted a+ib complex exponent of R and that x [1, 2] is a restricted complex power if

1Some version of these results has previously appeared in Tychonievich [33].

16 a+ib and only if the restriction x [1, 2] : [1, 2] → C is definable in R (the interval [1,2] + is chosen for the sake of convenience; any compact infinite subinterval of R would work just as well). We write E(R) to denote the set of restricted complex exponents for R, or just E if R is clear from context. Whenever convenient, a restricted complex power (or any partially defined function) can be extended to be totally defined by setting it equal to 0 off its domain.

a a+ib ib a+ib a For a, b ∈ R, notice that x = |x | and x = x /x , so a + ib ∈ E if and only if both a ∈ E and ib ∈ E. Letting K = E ∩ R be the set of real elements of E, we see that K is a subfield of R and that E is a vector space over K. Note that E = C if both exp [0, 1] and sin [0, 1] are definable, and that E ⊆ R if one of the functions exp [0, 1] or sin [0, 1] is definable but not the other (in fact, E = R if exp [0, 1] is definable and sin [0, 1] is not). Lemma 3.4.1 and Proposition 3.4.3 characterize structures for which E is a field.

As our main technical results, we calculate E for certain expansions of R. The first of these results is then applied to solve (Theorem 3.3.1) a problem of Miller [23] for expansions of R by locally closed trajectories of linear vector fields. To state these results, we define an operation on subsets of C as follows. Fix Z ⊆ C. Let X =

{Re z : z ∈ Z} ∪ {1} and Y = {i Im z : z ∈ Z}. Let F be the subﬁeld of R generated by X and the set {a/b : b 6= 0 and a, b ∈ SpanQ(X) Y }. Put V (Z) = F + SpanF (Y ). Then

z Theorem 3.1.1. For any Z ⊆ C, the structure (R, (x [1, 2])z∈Z ) has set of restricted complex exponents V (Z).

z Theorem 3.1.2. For any Z ⊆ C, the structure (R, sin [0, 1], (x [1, 2])z∈Z ) has set of restricted complex exponents V (Z) if Z ⊆ R, and has set of restricted complex exponents C otherwise.

For Z ⊆ R, Theorem 3.1.1 is immediate from a result of Bianconi [3]: The set 17 z of restricted real exponents of the structure (R, (x )z∈Z ) is Q(Z) if Z ⊆ R. Our proof for Theorem 3.1.1 for complex powers is along similar lines as his proof for real powers. We omit the proof of Theorem 3.1.2, as it is a routine modiﬁcation of that for Theorem 3.1.1.

Later in the chapter, we demonstrate conditions under which E(R) and E(R0) are isomorphic for elementarily equivalent structures R and R0 (Theorem 3.5.4). The chapter closes with some generalizations, open problems, and a sketch of how one might treat structures over ﬁelds other than R.

3.2 Proof of main result

Before proving Theorem 3.1.1, we ﬁrst need a few facts regarding the algebraic structure of E and the behavior of V.

Lemma 3.2.1. Let R be an expansion of R. The quotient of any two nonzero imaginary elements of E is an element of E. If E *R, then E is a vector space of dimension 2 over the ﬁeld E ∩ R.

1 Proof. For a nonzero real number a, let ga : [1, exp(2π/|a|)) → S be given by ga(x) = ia 1 2 x , where S denotes the unit circle in R . Observe that ga is a bijection and is ia definable if and only if x [1, 2] is definable. Suppose a, b ∈ E ∩ R are nonzero. −1 Then the function f : [1, exp(2π/|a|)) → [1, exp(2π/|b|)) given by f = gb ◦ ga is definable and bijective. A calculation then shows that f(x) = xa/b on the interval

a/b [1, exp(2π/|a|)). If [1, 2] ⊆ [1, exp(2π/|a|)), then f[1, 2] is the function x [1, 2]. If not, apply the multiplicative property of powers to deﬁnably extend f to the interval

[1, 2]. That E is a vector space of dimension 2 over E ∩ R now follows.

Lemma 3.2.2. The operation Z 7→ V (Z) is an abstract closure operation on subsets

0 0 of C. That is, for all Z,Z ∈ P(C) such that Z ⊆ Z , we have Z ⊆ V (Z), V (V (Z)) = 18 0 V (Z), and V (Z) ⊆ V (Z ). A set Z is V -closed if and only if Z ∩ R is a ﬁeld and Z is a vector space over Z ∩ R of either dimension 1 or 2. For any expansion R of R, E(R) is V -closed.

0 0 Proof. Let Z,Z ⊆ C. It is clear that Z ⊆ V (Z) and that Z ⊆ Z implies V (Z) ⊆ 0 V (Z ). We have shown that V (Z)∩R is a ﬁeld, V (Z) is a vector space over V (Z)∩R, and that V (Z) is closed under taking quotients of nonzero imaginary elements (and hence always of dimension 1 or 2). It is clear from construction that V ﬁxes exactly the sets satisfying these properties.

We are now ready for the

z Proof of Theorem 3.1.1. Let Z ⊆ C and R = (R, (x [1, 2])z∈Z ). We show that V (Z) = E. We have that V (Z) ⊆ E by Lemma 3.2.2 and that Z ⊆ E. For the opposite inclusion we give details only for the special case Z = {iα, iβ}, where the set {α, β} ⊆ R is algebraically independent. We do this to avoid diﬃculties due solely to notation; the general case is obtained by clerical modiﬁcation.

Suppose toward a contradiction that V (Z) 6= E, and let a, b ∈ R be such that a+ib a+ib a + ib ∈ E \ V (Z) and x [1, 2] is definable. Since x [1, 2] is definable if and a ib only if x [1, 2] and x [1, 2] are definable, we assume without loss of generality that ib b/α either a = 0 or b = 0. By Lemma 3.2.1, x [1, 2] is definable if and only if x [1, 2] is definable, so we assume without loss of generality that b = 0.

It is convenient from a technical standpoint to work with a structure interdeﬁnable with R in which the function symbols represent functions that are real analytic on

R. For ω ∈ R, deﬁne the functions fω, fiω, cω, sω : R → C as follows:

2x2 + 1 2x2 + 1 c (x) = cos ω log , s (x) = sin ω log ω x2 + 1 ω x2 + 1

2x2 + 1ω f (x) = , f (x) = c (x) + is (x) ω x2 + 1 iω ω ω 19 0 0 Let R = (R, cα, sα, cβ, sβ, α, β). That R is interdeﬁnable with R is clear, and the functions cα, sα, cβ, sβ : R → R are real analytic. By van den Dries [7], R is o-minimal, hence so is R0. The structure R0 is both existentially and universally interdeﬁnable with the structure

(R, exp(α arctan(x)), exp(β arctan(x)), α, β), which is model complete by Wilkie [34, First Main Theorem], so R0 is model complete.

a Since x [1, 2] is definable in R, we have that fa is definable in R and hence definable in R0. By basic facts about real closed fields, there is a term F (x, y, w) such that y = fa(x) if and only if ∃w(F (x, y, w) = 0). By Wilkie desingularization [34, Corollary

0 3.2], there are positive integers p, q and terms gk,j in the language of R in which the functions fiα and fiβ are only applied to variables, such that y = fa(x) if and only if

p q _ ^ ∂(g0,j, . . . , gq,j) ∃w gk,j(x, y, w) = 0 ∧ det 6= 0 . ∂(y, w1, . . . , wq) j=1 k=0

Without loss of generality, we assume that this q is minimal over all such nonsingular systems, even if terms from (R, (fz)z∈V (Z), (z)z∈V (Z)) are allowed. Then q > 0 since a∈ / V (Z). We assume that fa is deﬁned at 0 by the ﬁrst system of equations, that is y = fa(0) if and only if

q ^ ∂(g0,1, . . . , gq,1) ∃w gk,1(0, y, w) = 0 ∧ det (0, y, w) 6= 0 . ∂(y, w1, . . . , wq) k=0

By o-minimality and the Implicit Function Theorem, this formula deﬁnes fa(x) for x in an open interval I containing 0 such that there are real analytic functions w1, . . . , wq with q ^ gk,1(x, fa(x), w1(x), . . . , wq(x)) = 0 k=0 for all x ∈ I. Notice that gk,1(x, y, w) is a polynomial in terms z, cα(z), sα(z), cβ(z), and sβ(z), where z can be any unary term occurring in x, y, or w. Let C be the ring

20 generated by series of the form z, sα(z), sβ(z) for z = x, fa(x), w1(x), . . . , wq(x). Since cω and sω are algebraically dependent, the above shows that

trdegC(C) < 3(q + 2) − (q + 1) = 2q + 5.

The remainder of the proof consists of deriving a contradictory lower bound for 2 2 ¯ trdegC(C). For ease of notation, putw ¯0(x) = (2x + 1)/(x + 1), fa =w ¯0 ◦ fa andw ¯j =w ¯0 ◦ wj for j > 0. Consider the set W made up of the formal series ¯ ¯ ¯ ¯ ¯ ¯ log fa(x)−log fa(0), iα(log fa(x)−log fa(0)), and iβ(log fa(x)−log fa(0)), along with the series

• logw ¯j(x) − logw ¯j(0), j = 0, . . . , q

• iα(logw ¯j(x) − logw ¯j(0)), j = 0, . . . , q

• iβ(logw ¯j(x) − logw ¯j(0)), j = 0, . . . , q

We assume that for each j we have wj(0) > 0. Each of the series in W has zero constant term, and each series converges on a neighborhood of zero.

Suppose toward a contradiction that the set W is Q-linearly dependent. Each element of W is either real or purely imaginary, so any Q-linear dependence relation that holds among the elements of W must hold among the real elements and the imaginary elements of W separately. Suppose that the set of imaginary elements 0 ¯ of W is Q-linearly dependent. Then the set of formal series W := {log fa(x) − ¯ log fa(0); logw ¯j(x) − logw ¯j(0) : j = 0, . . . , q} is Q(α/β)-linearly dependent. The linear dependence relation witnessing this must involve nontrivially at least one of the terms logw ¯j(x) − logw ¯j(0), j ≥ 1, and without loss of generality we take j = q. Solving this dependence relation for logw ¯q(x) and exponentiating, we present ¯ the functionw ¯q(x) asw ¯q(x) = c · h(fa(x), w¯0(x),..., w¯q−1(x)), where c ∈ R and the function h(¯u, w¯0,..., w¯q−1) is a product of the termsu, ¯ w¯0,..., w¯q−1 raised to 21 powers from Q(α/β). We see that h(¯u, w¯0,..., w¯q−1) is a term in the language 0 of R , and that cα(z), sα(z), cβ(z), and sβ(z), z = h(¯u, w¯0,..., w¯q−1), are equal to

0 0 polynomials in terms (fζ (z ))ζ∈V (Z), where z ranges over u and the unary terms occurring in w (as before we denotex ¯ = w0(x)). Thus, we may replace wq by

−1 w¯0 (c · h(¯u, w¯0,..., w¯q−1)) in the system of equations deﬁning fa. Expand the deter-

th minant ∂(g0,1, . . . , gq,1)/∂(y, w1, . . . , wq) by minors along the q column. Since

∂(g0,1, . . . , gq,1) (fa(0), w0(0), . . . , wq(0)) 6= 0, ∂(y, w1, . . . , wq) we have that one of these minors is nonsingular at 0, and hence in a neighborhood

0 of 0. Delete the corresponding equation and note that wq(0) 6= 0; it is a Calculus problem to check that the resulting system is nonsingular, a contradiction. Thus the set of imaginary elements of W is Q-linearly independent. A nontrivial Q-linear dependence relation among the real elements of W would yield a nontrivial Q(α/β)- 0 linear dependence relation on W , so W itself is Q-linearly independent. By Ax [1], we have

w trdegC(C[W ∪ {e : w ∈ W }]) ≥ 1 + 3(q + 2) = 3q + 7.

The set W is C-linearly dependent, however, and hence algebraically dependent over

C. Since dimC(W ) ≤ q + 1, the diﬀerence between the transcendence degrees of w w C[W ∪ {e : w ∈ W }] and C[{e : w ∈ W }] is at most q + 1. Thus,

w trdegC(C[{e : w ∈ W }]) ≥ 3q + 7 − (q + 1) = 2q + 6.

w The rings C[{e : w ∈ W }] and C have the same transcendence degree over C, so

2q + 6 ≤ trdegC(C) ≤ 2q + 5.

This is a contradiction, so fa is not deﬁnable, and hence E = V (Z).

22 Unbounded domains

Theorems 3.1.1 and 3.1.2 were stated and proved for powers deﬁned on bounded intervals, but in some cases we are able to extend the domains of the powers considered to unbounded intervals while still maintaining those theorems’ conclusions. For real powers, we do this by modifying the proof of Theorem 3.1.1 in a straightforward

0 manner (see Bianconi [3]) to see that for Z ⊆ C and Z ⊆ Z ∩ R, the structure z r (R, (x [1, 2])z∈Z , (x )r∈Z0 ) has set of restricted complex exponents V (Z). For imaginary powers, we see below that the situation is more interesting.

iω Proposition 3.2.3. Let Z ⊆ C and let ω ∈ R such that iω ∈ Z. Let R = (R, x , z (x [1, 2])z∈Z ). Then E(R) = V (Z).

Proof. By Miller [21, Theorem 4], if a function is deﬁnable on the interval [1, 2] in R, then a restriction of this function to some open subinterval of [1, 2] is deﬁnable in the

iω z structure (R, x [1, 2], (x [1, 2])z∈Z ). Apply Theorem 3.1.1 to this structure, noting that iω ∈ Z.

Even though E behaves nicely for an expansion by real powers or a single imaginary power, the case of two or more imaginary exponents is very diﬀerent. To put the following result in context, recall from the previous chapter that the sets deﬁnable in the structure (R, Z) form what is called the real projective hierarchy, which considered wild from a model theoretic point of view (again, see Kechris [17]). For any

n n ∈ N, each Borel subset of R is deﬁnable in (R, Z), and as a consequence, global complex exponentiation is deﬁnable in (R, Z). In particular, E((R, Z)) = C.

iα iβ Proposition 3.2.4. Let α, β ∈ R be Q-linearly independent, and let R = (R, x , x ).

Then Z is deﬁnable in R.

iω + Proof. For any nonzero real ω, the set Gω := {x > 0 : x ∈ R } is a cyclic multiplicative subgroup of R, hence discrete. 23 + As α and β are Q-linearly independent, Gα ·Gβ is dense in R . By Hieronymi [16],

Z is deﬁnable.

3.3 Linear vector ﬁelds

We now apply Theorem 3.1.1 to the study of expansions of R by trajectories of linear vector ﬁelds as in [23]. Before this, we require a few deﬁnitions. A linear vector

n n field is an R-linear transformation F : R → R for some n ≥ 1, and a solution of F n is a differentiable function γ : I → R defined on a nontrivial interval I ⊆ R such that γ0(t) = F (γ(t)) for all t ∈ I.A trajectory of F is the image of a solution, viewed as a set with neither parameterization nor orientation. A trajectory is locally closed if it is the intersection of an open set and a closed set. The following is a classification, up to interdefinability, of expansions of R by families of locally closed trajectories of linear vector fields.

Theorem 3.3.1. Let G be a collection of locally closed trajectories of linear vector

ﬁelds. Let Z denote the image of Z 7→ V (Z): P(C) → P(C).

1. If (R, (Γ )Γ ∈G) does not deﬁne exp [0, 1], then it is interdeﬁnable with exactly one of the following:

a z • (R, (x )a∈K , (x [1, 2])z∈Z ) for some subﬁeld K of R such that K ⊆ Z ∈ Z

ib z • (R, x , (x [1, 2])z∈Z ) for some nonzero b ∈ R such that ib ∈ Z ∈ Z

2. If (R, (Γ )Γ ∈G) defines exp [0, 1], but neither sin [0, 1] nor exp, then it is inter- a definable with (R, exp [0, 1], (x )a∈K ) for some subfield K of R

3. If (R, (Γ )Γ ∈G) deﬁnes both exp [0, 1] and sin [0, 1], but not exp, then it is in- terdeﬁnable with exactly one of the following:

24 RE a • (R , (x )a∈K ) for some subﬁeld K of R

RE ib • (R , x ) for some nonzero b ∈ R

4. If (R, (Γ )Γ ∈G) deﬁnes exp, then it is interdeﬁnable with exactly one of the following:

• (R, exp)

• (R, exp, sin [0, 1])

• (R, Z).

Proof. A locally closed trajectory Γ of a linear vector field either has compact closure or infinite logarithmic length. Write G = H ∪ K, where every Γ ∈ H has infinite logarithmic length and every Γ ∈ K has compact closure. By [23], (R, (Γ )Γ ∈H) is interdefinable with exactly one of the following:

r • (R, (x )r∈K ) for some subﬁeld K of R

ib • (R, x ) for some nonzero real b

• (R, exp)

• (R, Z).

An examination of the proof in [23] shows that (R, (Γ )Γ ∈K) is interdeﬁnable with at least one of the following:

z • (R, (x [1, 2])z∈Z ) for some Z ∈ Z

• (R, exp [0, 1])

RE • R

25 RE By Bianconi [2], R and (R, exp [0, 1]) are not interdeﬁnable, and by Theorem 3.1.1, w there is no Z ∈ Z such that (R, (x [1, 2])z∈Z ) is interdeﬁnable with (R, exp [0, 1]) RE or R . As (R, (Γ )Γ ∈G) can be written as the expansion (R, (Γ )Γ ∈H, (Γ )Γ ∈K), and so

(R, (Γ )Γ ∈G) is interdefinable with an expansion of one of the structures on the first list by the definable functions of one of the structures on the second list. The claim follows from basic properties of restricted complex powers, Proposition 3.2.3 and the comments immediately preceding it, and a call to Miller [20].

Remarks. (i) The classes of structures given in the statement of Theorem 3.3.1 are distinct by Theorem 3.1.1 and Bianconi [2,3].

(ii) As mentioned by Miller [23, Section 3], after suitable normalization it is possible to calculate the ﬁeld K and the set Z (when appropriate) in terms of the ratios of eigenvalues of the vector ﬁelds of which the sets in G are trajectories.

(iii) As in [23], Theorem 3.3.1 may be extended expansions of Re by families of trajectories that are not all closed through the use of Corollary 2.2.1 from the previous chapter.

3.4 Complex powers on subsets of C

We proceed to study the situation of complex powers defined on open subsets of C. Bianconi [3] showed that any holomorphic (complex differentiable) function definable

z on an open subset of C in the structure (R, (x )z∈Z ), where Z ⊆ R, is deﬁnable in

R. Thus, in an expansion of R by real power functions, all holomorphic deﬁnable functions are semialgebraic. This is not always the case when restricted imaginary powers are involved.

Lemma 3.4.1. Let R be an expansion of R. Let Y be the set of imaginary elements of E(R). The following are then equivalent:

26 • The set E is a ﬁeld.

• The set Y \{0} is closed under the taking of multiplicative inverses.

• The square of each element of Y lies in E.

• Either Y = {0} or there are two nonzero elements of Y such that their product

lies in E.

Proof. This equivalence is clear if E ⊆ R. Otherwise, let iα ∈ Y be nonzero and put

K = E ∩ R. Notice that E = K + iαK is a vector space of dimension 2 over K. Mul- tiplication by a nonzero element of Y induces a nonsingular linear transformation on

C. If one such linear transformation preserves E, then all such linear transformations preserve E. This shows that the conditions listed above are equivalent.

z Proposition 3.4.2. Let Z ⊆ C and let R = (R, (x [1, 2])z∈Z ).

1. If there exists ω ∈ R such that V (Z) contains both iω and i/ω, then the function iω x [1, 2] extends deﬁnably to a holomorphic function on some nonempty open subset of C.

2. If V (Z) contains a nonzero imaginary element, then for any real ω ∈ V (Z),

ω the function x [1, 2] extends deﬁnably to a holomorphic function on some nonempty open subset of C.

1 Proof. For (1), suppose that ω > 0 and let f : [1, exp(π/ω)) → S be given by iω f(x) = x [1, exp(π/ω)). Notice that f is a bijection, its inverse is a branch of the 1/iω −i/ω 1 analytic function z = x on S , and f is definable if and only if iω ∈ V (Z). Suppose iω ∈ V (Z) and let S = {reiθ : 1 ≤ r < exp(π/ω), 0 ≤ θ < π}. We define the function f˜ : S → C by f(|z|) f˜(z) = f −1(z/|z|) 27 ˜ iω ˜ Observe that f is an analytic continuation of x [1, exp(π/ω)) to S and that f is definable.

The proof of (2) is similar: given ia ∈ V (Z) with a ∈ R nonzero, conjugate ω ia ω x [1, 2] by x [1, 2] to deﬁne z on a nontrivial arc of the unit circle. Then proceed as above to deﬁne zω on a nonempty open subset of the plane.

Remark.

If R is o-minimal, then by cell decomposition, algebraic properties of power functions, and some basic results in complex analysis, the preceding proof can be extended to allow S to be any bounded deﬁnable subset of C which lies within some compact, 2 simply connected subset of R \{0}. Combining Lemmas 3.2.2 and 3.4.1 with Proposition 3.4.2, we have the following characterization.

Proposition 3.4.3. The set E is a field if and only if either E ⊆ R or each restricted complex power definable on a nonempty open interval extends definably to a holomorphic function on a nonempty open subset of the plane.

Combined with Proposition 3.2.3, this yields a curious dichotomy:

iω Corollary 3.4.4. Let ω ∈ R. A branch of z is deﬁnable on some nonempty open iω subset of C in the structure (R, x ) if and only if ω is either rational or quadratic irrational.

Proof. By Proposition 3.2.3, the set of restricted complex exponents for the structure

iω (R, x ) is V ({iω}) = Q + iωQ, which is a ﬁeld if and only if ω is either rational or quadratic irrational. Apply Proposition 3.4.3.

28 3.5 Invariance of E under elementary equivalence

Fix an expansion R of R. We show that E(M) and E(R) are isomorphic when R is an o-minimal expansion of R that does not deﬁne exp [0, 1] and M is elementarily equivalent to R. We start with a

+ Lemma 3.5.1. Let I ⊆ R be a nonempty open interval. The following are equivalent:

2 1. R deﬁnes exp zI

iy 2 2. R deﬁnes x I

3. R deﬁnes exp I and sin I

Proof. The equivalence (1) ⇔ (3) is clear, so we consider the implication (2) ⇒ (3).

Since xiy = (cos(y log x), sin(y log x)), this follows via partial derivatives.

With this, we turn our attention to definable families of functions. We show that a definable family of functions can contain only finitely many restricted complex powers unless there is a nonempty open interval such that exp I is definable.

Proposition 3.5.2. Suppose that no restriction of exp z to a nonempty open subset of

n C is deﬁnable. Let A ⊆ R , let I be a nonempty open interval, and let f : A × I → R be deﬁnable. Then the set

iω Ω := {ω ∈ R : ∃a ∈ A, ∀x ∈ I, f(a, x) = x } is ﬁnite. If R does not deﬁne exp J for any nonempty open interval J, then the set

b B := {b ∈ R : ∃a ∈ A, ∀x ∈ I, f(a, x) = x } is ﬁnite.

29 Proof. Suppose toward a contradiction that Ω is inﬁnite. By o-minimality, Ω must

iy 2 contain a nonempty open subinterval J. Thus the map x : I × J → R is definable, a contradiction. The proof that B is finite if R does not define exp J is similar.

Corollary 3.5.3. 1. Suppose that no restriction of exp z to a nonempty open sub-

set of C is definable and that ω ∈ R is such that iω ∈ E. Let I be a nonempty, iω open, ∅-definable interval such that x I is definable. Then the restricted power iω x I is ∅-definable.

2. Suppose that exp J is not definable for any open interval J and that b ∈ R is b such that b ∈ E. Let I be a nonempty, open, ∅-definable interval such that x I b is definable. Then the restricted power x I is ∅-definable.

iω Proof. For (1), suppose that x I is not ∅-definable. By standard o-minimal argu- iω ments (see [19], for instance) there is a function f : A×I → R such that f(a, x) = x α for x ∈ I. The set {α ∈ R : ∃a ∈ A : f(a, x) = x for x ∈ I} is a definable subset of R, and so must be a finite union of intervals. If one of these intervals is infinite, then the set Ω as defined in Proposition 3.5.2 is infinite. Thus A must be a finite set,

iω a contradiction to the assumption that x I is not ∅-deﬁnable. The proof for (2) is similar.

Subject to the condition that exp J is not deﬁnable for any nonempty open interval J, the proof of Corollary 3.5.3 shows that deﬁnable restricted complex powers

z x I are ∅-definable up to parameters used to define I. Over R we can always take [1, 2] ⊆ I, so all restricted complex powers are actually ∅-definable.

To state the final theorem of this section, we need a more general definition of the set of restricted complex exponents. Let R be an o-minimal expansion of a real closed field with underlying set R, and write C to mean the algebraic closure of R considered as a definable object in R.A restricted complex power is a partial 30 multiplicative homomorphism f : I → C such that I ⊆ R is a nontrivial interval and 1 ∈ I. By o-minimality, f is differentiable in a neighborhood of 1. We say that the restricted complex exponent of f is the value f 0(1), and we define the set of restricted complex exponents E in the obvious way. These definitions are equivalent to our earlier definitions if R = R. If R is elementarily equivalent to an o-minimal expansion of R, then the results of this section apply with no additional argument required, and we have

Theorem 3.5.4. Suppose R is an o-minimal expansion of R that does not define exp [0, 1] and suppose that M is elementarily equivalent to R. Let M be the underlying set of M. Then M ∩ E(M) and R ∩ E(R) are isomorphic as ordered fields. If E(R) is a field, then the isomorphism extends to a field isomorphism of E(M) and

E(R). Otherwise, the isomorphism extends to an isomorphism of E(M) and E(R) as vector spaces over R ∩ E(R).

3.6 Optimality

O-minimal expansions of real closed ﬁelds

Many of the results in this chapter can be stated and proved for o-minimal expansions of real closed fields. Let R be an o-minimal expansion of a real closed field with underlying set R, and write C to mean the algebraic closure of R considered as a definable object in R. The proofs of the results regarding the behavior of V on subsets of C and the definability of powers on open subsets of C carry over in an obvious way, while our proof for Theorem 3.1.1 depended on results from Wilkie [34] that only apply to expansions of R (Foster [13] has a generalization of the results needed from [34]). The invariance results for E generalize with a little work. If R supports abstract exponential and arctangent functions, that is, nonconstant definable functions f, g :

31 R → R such that f(x + y) = f(x)f(y) and g(x) + g(1/x) = c ∈ R, then the proofs given above work modulo an argument at the end involving quantiﬁcation over parameters and the existence of prime models for o-minimal structures (Pillay and Steinhorn [27]). If not, then a more delicate treatment as done by Miller [20] can be used.

Other expansions of the real ﬁeld

For at least one other kind of expansion of o-minimal expansion of R, Theorem 3.1.1 generalizes. Let R be an expansion of R, and for z ∈ R, put

gz(x, y) := (cos(z arccos x), sin(z arcsin y)).

0 Let E (R) be the set of all z ∈ R such that there is a nontrivial arc S of the unit circle and the restriction gzS is deﬁnable in R. If a nontrivial restricted imaginary power is deﬁnable, then an examination of the proof of Proposition 3.4.2 shows that

E0 = E. Otherwise, argument along the lines of our proof of Theorem 3.1.1 yields the following result:

0 iθ Proposition 3.6.1. Let Z,Z ⊆ R, and put S = {e : 0 ≤ θ < π/4} ⊆ C. Then

0 0 • E ((R, exp [0, 1], gz0 S)z0∈Z0 )) = V (Z )

0 z 0 • E ((R, (x [1, 2])z∈Z , (gz0 S)z0∈Z0 )) = V (Z )

z • E((R, (x [1, 2])z∈Z , (gz0 S)z0∈Z0 )) = V (Z)

How else might Theorem 3.1.1 generalize? Let R be an o-minimal expansion of R such that neither sin [0, 1] nor exp [0, 1] is deﬁnable. Though the structure z (R, (x [1, 2])z∈Z ) is known to be o-minimal and exponentially bounded as a conse- z quence of Speissegger [30], it is not known at this time if (R, (x [1, 2])z∈Z ) must be

32 polynomially bounded, let alone what its set of restricted complex exponents is. In

z fact, it is not known if E((R, (x [1, 2])z∈Z )) = V (E(R) ∪ Z) holds, even assuming z that E((R, (x [1, 2])z∈Z )) 6= C and that R is an expansion of R by restrictions of analytic functions.

33 CHAPTER 4

METRIC PROPERTIES OF SETS DEFINABLE IN (R, α−N)

4.1 Introduction

Let R be an o-minimal expansion of the real ordered ﬁeld R having ﬁeld of exponents

Q and let 1 < α ∈ R. In this chapter we prove some results about the structure (R, α−N). This builds on the pioneering work of van den Dries [6] and Miller [19,21] that produced quantiﬁer elimination and tameness results for nontrivial, and even non-o-minimal, expansions of R. Our main result is a version of of Theorem 1.3.2 for the structure (R, α−N):

n 1 Theorem 4.1.1. Let M ⊆ R be a nonempty, bounded, d-dimensional, C -submanifold n − of R that is deﬁnable in (R, α N). Then

dim(fr M) < d if and only if vold(M) < ∞.

We do not require submanifolds to be connected. The proof requires a number of technical preliminaries, beginning with a calculation of structure induced on α−N in

(R, α−N). The structure induced on α−N in (R, α−N) is the structure over α−N consisting of subsets of Cartesian powers of α−N deﬁnable in the structure (R, α−N).

Theorem 4.1.2. The structure induced on α−N in (R, α−N) is interdeﬁnable with the structure (α−N, · , <).

34 To prove this technical result, we will use results due to Cluckers [4] about the sets deﬁnable in models of the theory of the ordered additive group of integers in the language {+, <, {≡ mod n}n∈N, 0, 1}, where the binary relation ≡ mod n denotes congruence modulo n as usual. We will call this theory the theory of Presburger

m arithmetic. For a set X ⊆ Z and a function f : X → Z, Cluckers defines the function to be linear if there is a constant γ ∈ N and integers ai, 0 ≤ ci < ni for m i ∈ {1, 2, . . . , m} such that for all x = (x1, x2, . . . , xm) ∈ Z we have that xi ≡ ci mod ni and m X ai f(x) = γ + (x − c ). n i i i=1 i Cells for expansions of Presburger arithmetic are defined inductively by Cluckers along similar lines to the definition of cells in an o-minimal structure. We say that an singleton is a Presburger 0-cell in Z. We define a Presburger 1-cell in Z to be any infinite subset of Z of the form

{x ∈ Z : α < x < β and x ≡ c mod n} where c, n ∈ N such that 0 ≤ c < n, either α ∈ Z or α = −∞, and either β ∈ Z or m β = ∞. Supposing that we have deﬁned what a Presburger k-cell in Z is, we deﬁne m+1 m+1 Presburger cells in Z as follows. A Presburger k-cell in Z is any set of the form

m+1 {(x, t) ∈ Z : x ∈ D, α(x) = t, and t ≡ c mod n}

m where D is a Presburger k-cell in Z , the numbers c, n ∈ N are such that 0 ≤ c < n, and the function α : D → Z is linear in the sense of Cluckers. A Presburger (k+1)-cell m+1 in Z is any set of the form

m+1 A = {(x, t) ∈ Z : x ∈ D, α(x) < t < β(x), and t ≡ c mod n}

m where D is a Presburger k-cell in Z , the numbers c, n ∈ N are such that 0 ≤ c < n, the function α : D → Z is either linear in the sense of Cluckers or always −∞, the 35 function β : D → Z is either linear in the sense of Cluckers or always ∞, and there is no finite uniform bound on the cardinality of the fibers Ax ⊆ Z (so that no (k+1)-cell is a finite union of k-cells).

m Theorem 4.1.3 (Cluckers). Let the set X ⊆ Z and the function f : X → Z be deﬁnable in the structure (Z, +, <, {≡ mod n}n∈N, 0, 1). Then there is a partition of

X into Presburger cells such that on each cell C, the restriction fC is linear in the sense of Cluckers.

There is a notion of minimality for expansions of models of the theory of Pres- burger arithmetic, similar to o-minimality. Let T be a complete theory extending

Presburger arithmetic, possibly in an expanded language. Given a model M of T , the structure (Z, <, +, 0, 1) embeds into (M, <, +, 0, 1) in the obvious way; we regard

Z as a subset of M via this embedding. We say that M is Presburger-minimal if every subset of Z deﬁnable in M is deﬁnable in (Z, <, +, 0, 1), and that T is Presburger- minimal if all of its models are Presburger-minimal.

Theorem 4.1.4 (Cluckers). With T as above, the following are equivalent:

• T is Presburger-minimal.

• Some model of T is Presburger-minimal.

n • Every subset of Z deﬁnable in any M |= T is deﬁnable in (Z, <, +, 0, 1).

Model completeness for (R, α−N)

We now prove a technical result about the structure (R, α−N). Henceforth, we will write H = α−N to simplify notation and eliminate ambiguities when writing Cartesian powers of α−N. We will assume that the language L of R has no relation symbols

36 except <, and that every definable function in the structure R is given by a function symbol. We lose no generality in these assumptions, as any relation may be replaced by its indicator function and the following results are only concerned with definability up to definability in R. Since o-minimal structures have definable Skolem functions, we see easily that under these assumptions the theory of R is universally axiomatizable and that it admits quantifier elimination.

To analyze the sets definable in (R,H), it is convenient to temporarily look at it as a structure in a slightly different language. Define a function Λ : R → R as follows:   0, if x ≤ 0 Λ(x) :=  n n n+1 α , if n ∈ Z and α ≤ x < α

Let LΛ be the expansion of the language L by a symbol for Λ. As

H = {x ∈ R : 0 < x ≤ 0 and Λ(x) = x} and

Λ(x) = max{y ∈ αZ ∪ {0} : y ≤ x < αy}, we see immediately that the structures (R, Λ) and (R,H) are interdeﬁnable. The next lemma shows how one may express LΛ-terms by formulas in the language L.

Lemma 4.1.5. For any LΛ-term t(x), there is a k ∈ N and there are L-terms n k s, s1, . . . , sk such that for all x ∈ R , there is a y ∈ H such that t(x) = s(x, y) and such that for i = 1, . . . , k we have that si(x, y1, . . . , yi−1) < yi ≤ αsi(x, y1, . . . , yi−1)

(i.e. yi = αΛ(si(x, y1, . . . , yi−1))).

0 0 Proof. By induction on LΛ-terms. Observe that if t is of the form Λ(t ), where t is an L-term, then t = s(y) := y if and only if y ∈ H and t0 < y ≤ αt0.

We combine this with the following syntactic result:

Theorem 4.1.6 (Miller [19]). The theory of (R, Λ) admits quantiﬁer elimination and is universally axiomatizable. 37 Together, these results give us a very strong form of model completeness for

(R,H):

n Corollary 4.1.7. If A ⊆ R is deﬁnable in (R,H), then there exists k ∈ N and an (n + k)-ary set B deﬁnable in R such that x ∈ A if and only if there exists y ∈ Hk such that (x, y) ∈ B.

Proof. By Theorem 4.1.6, any deﬁnable function in (R, Λ) is given piecewise by terms, so we may assume that there is an n-ary term t(x) such that x ∈ A iﬀ t(x) = 1. By

Lemma 4.1.5, there is a k ∈ N and there are L-terms s, s1, . . . , sk such that for all n k x ∈ R , there is a y ∈ H such that t(x) = s(x, y) and such that for i = 1, . . . , k we n+m have that si(x, y1, . . . , yi−1) < yi ≤ αsi(x, y1, . . . , yi−1). Let B ⊆ R be deﬁned by the formula

k ^ s(x, y) = 1 ∧ si(x, y1, . . . , yi−1) < yi ≤ αsi(x, y1, . . . , yi−1) . i=1

Then B is deﬁnable in R and x ∈ A if and only if x satisﬁes (∃y ∈ Hk)(x, y) ∈ B.

Trivialization and deﬁnable sections

We now make use of several structure theorems for sets deﬁnable in o-minimal structures to extract information from Corollary 4.1.7.

First, let us discuss definable trivialization for functions definable in an o-minimal structure. Fix an o-minimal structure M. Let A ⊆ M m and S ⊆ M n be definable sets and let f : S → A be a definable function. We view f as parameterizing the

−1 deﬁnable family of sets (f ({a}))a∈A, where A is the parameter space. A deﬁnable

N trivialization for f is a pair (X, λ) such that there is an N ∈ N such that X ⊆ M is a definable set, λ : S → X is a definable function, and the function (f, λ): S → A×X is a homeomorphism. The function f is said to be definably trivial if it has a definable

38 trivialization, and f is said to be deﬁnably trivial over A0 ⊆ A if the restriction

−1 0 ff (A ) is deﬁnably trivial.

Theorem 4.1.8 (Deﬁnable Trivialization [9]). Let R be an o-minimal expansion of

m n R. Let the sets A ⊆ R and S ⊆ R be definable and let the function f : S → A be definable. Then there are finitely many disjoint definable cells A1,...,AN such that

A = A1 ∪ ... ∪ AN and f is deﬁnably trivial on Ai for each i.

N In an o-minimal expansion of R, the set R is definably homeomorphic to any N N-simplex σ ⊆ R . Thus we have that for any definable set X and for any N-simplex N N σ ⊆ R , that X × R is definably homeomorphic to X × σ. We say that a definable section is a definable continuous function F : X × Z →

n+m n m R such that the sets X ⊆ R and Z ⊆ R , for each z ∈ Z and x ∈ X we have π2(F (x, z)) = z, where π2 is projection on the last m coordinates. We say F is injective if F is injective as a function, or equivalently if for each z ∈ Z the function

Fz : x 7→ π1(F (x, z)) is injective, where π1 is projection on the ﬁrst n coordinates. A

n [ collection of sections F covers the set Y ⊆ R if Y = {π1(Image(F )) : F ∈ F}.

n+m n Proposition 4.1.9. Let S ⊆ R be a definable set, let G : S → R be a definable function, and let p ∈ N. Then there is a finite collection of injective definable sections n F such that for each function Gz : x 7→ G(x, z), the image of the set Sz := {x ∈ R :

(x, z) ∈ S} is the union of the projections of Fz(Sz) on the ﬁrst n coordinates. The

m p domain of each F ∈ F is of the form σ × C where C ⊆ R is a C cell and σ is a [ simplex. The domains of these sections are pairwise disjoint, and the function F is injective.

p p Proof. By C cell decomposition, there is a ﬁnite decomposition of R into C cells

Si, i = 1, . . . , a, compatible with {S} such that the function (x, z) 7→ (G(x, z), z) is continuous on each Si, injective on the ﬁrst b cells, and G(S1 ∪ ... ∪ Sb) = G(S). 39 For a simplicial complex K, we denote by |K| the union of all simplices contained in K. By Theorem 4.1.8 and the o-minimal triangulation theorem [9], there is a

m p 0 further decomposition of R into C -cells Cj, j = 1, . . . , a , such that for each j there n n is a complex Kj in R and homeomorphisms hi,j : Si ∩ (R × Cj) → |Kj| × Cj such −1 that π2(hi,j(x, z)) = z. By taking the preimage hi,j (σ × Cj) for each σ ∈ Kj, we see

n p there is a decomposition of R by C cells Dk, k = 1, . . . , c, and for each triple i, j, k there is a simplex σi,j,k and a homeomorphism hi,j,k : Si ∩ (Dj × Ci) → σi,j × Ci such that π2(hi,j,k(x, z)) = z.

For each function Gz : x 7→ G(x, z), we thus have that Gz(Sz) is the union of all

−1 images G(hi,j,k(σk, z)) such that z ∈ Cj ∩ π2(Si) for some i ≤ b and some j ≤ c, and −1 on these sets each function x 7→ G(hi,j,k(x, z)) is injective.

If one thinks of G above as a deﬁnable family of sets (Gz(Sz)), this result shows that there is a ﬁnite family of injective sections F such that each set Gz(Sz) is the

n+m union of all π1 ◦ F (σF × Cz) for F ∈ F . Given any definable family S ⊆ R of n n subsets of R , the function G : S → R given by G(x, z) = x displays the sets Sz in this form, so any definable family of sets can be covered by finitely many injective sections. We apply this to (R,H):

n Theorem 4.1.10. Let A ⊆ R be a set deﬁnable in the structure (R,H) and let p ∈ N. Then there are ﬁnitely many continuous functions (fi)i∈I such that A is the

mi di disjoint union of the images fi(H × (0, 1) ) and

mi di n • Each function fi : (0, 1) × (0, 1) → R is deﬁnable in the structure R.

mi p • For each y ∈ (0, 1) , the function x 7→ fi(y, x) is C .

mi • For each y ∈ (0, 1) , the function x 7→ fi(y, x) is injective.

40 Proof. Lemma 4.1.7 implies that every definable set is a union over elements of Hm of a family of sets definable in R. We apply Proposition 4.1.9 to this family of definable sets to get the required family of functions.

Thus, to study nonempty sets A deﬁnable in (R,H), we may assume without loss

m d m d n of generality that A = f(H × (0, 1) ), where f : (0, 1) × (0, 1) → R is continuous and deﬁnable in R, d = dim A, and such that for each y ∈ (0, 1)m, the function x → f(y, x) is injective and of class Cp.

4.2 Induced structure

We are now ready for the

Proof of Theorem 4.1.2. We must show that the structure H induced on H in (R,H) is interdeﬁnable with (H, · , <). As H is an expansion of (H, · , <), it suﬃces by

Theorem 4.1.4 to show that every subset of H definable in (R,H) is actually definable in (H, · , <). Let A ⊆ H be definable in the structure (R,H). As dim A = 0, we may assume by Theorem 4.1.10 that A = f(Hm) for some m-ary function f definable in

R. Thus, it suﬃces to show that the set graph(f)∩Hm+1 is deﬁnable in the structure

(H, · , <), as this then implies that the image f(Hm) is deﬁnable in (H, · , <).

We now proceed by induction on m ≥ 0 to prove the somewhat stronger result

m m that for every set A ⊆ R deﬁnable in the structure R, the intersection A ∩ H is deﬁnable in the structure (H, · , <). The cases m = 0 and m = 1 are trivial, so

k let m > 0 be an integer such that for any integer 0 ≤ k < m and any set B ⊆ R k m definable in R, the set B ∩ H is definable in (H, · , <). Let A ⊆ R be definable in the structure R.

First suppose that 0 is not a limit point of A. Then there is an > 0 such that A

m i−1 m−i is disjoint from [0, ) . For i = 1, . . . , m, put Ri := (0, 1) × [, 1) × (0, 1) and

41 m m [ [ put R = Ri. Then A ⊆ R, so A = A ∩ Ri. We consider the summand A ∩ Rm. i=1 i=1 Since the intersection [, 1) ∩ H is finite, we have that A ∩ Rm is a finite union of sets of the form A ∩ (0, 1)m−1 × {h} for some h ∈ H. Each of these sets is definable in the structure (H, · , <) by the inductive hypothesis, so A ∩ Rm is definable in (H, · , <).

m A similar argument shows that for each i, A ∩ Ri is deﬁnable in (H, · , <), so A ∩ H is deﬁnable in (H, · , <).

For the remainder of this proof, we assume without loss of generality that A is a cell.

Case 1: A is the graph of a function. We now suppose that 0 is a limit point of A and that there is cell D ⊆ (0, 1)m−1 and a function f : D → (0, 1), deﬁnable in R, such that A is the graph of f. By the preparation theorem of van den Dries and Speissegger [12], we resolve D into cells deﬁnable in R, such that for each cell C

m−2 there are numbers λ, µ ∈ Q with µ > 0, λ ≥ 0 and functions θ, a, b : (0, 1) → R m−1 and u : (0, 1) → R, each deﬁnable in R, such that either θ = 0 or 0 < |y − θ(x)| < |θ(x)|/α2 for all (x, y) ∈ C, and such that

f(x, y) = |y − θ(x)|λa(x)u(x, y), and

|u(x, y) − 1| < min(α−2, |y − θ(x)|µ|b(x)|) for all (x, y) ∈ C.

Case 1.1: λ = 0. On a cell C such that λ = 0, we have that

f(x, y) = a(x)u(x, y) and |u(x, y) − 1| < min(α−2, |y − θ(x)|µ|b(x)|) for all (x, y) ∈ C. By the inequality |u(x, y) − 1| < α−2, we have for each (x, y) ∈

C ∩ Hm−1 that f(x, y) ∈ H implies that f(x, y) must be the closest element of H to 42 a(x). The set {(x, z) ∈ Hm : z is the closest element of H to a(x)} is the intersection of a tube around the graph of the function a with the set Hm−1, and hence it is definable in the structure (H, · , <) by the inductive hypothesis. In fact, it is the graph of a function g : Hm−2 → H definable in (H, · , <). By Theorem 4.1.3, this function is piecewise linear in the sense of Cluckers. Using the formula defining g as a piecewise linear function in the structure (H, · , <), we interpolate to a function g¯ : (0, 1)m−2 → (0, 1) definable in the structure R. The set B := {(x, y) ∈ C :g ¯(x) = f(x, y)} ∩ Hm is definable in (H, · , <) by the inductive hypothesis. As g is definable in (H, · , <), the set graph(f) ∩ Hm = {(x, y, g(x)) : (x, y) ∈ B} ∩ Hm is definable in the structure (H, · , <) as claimed.

Case 1.2: λ > 0, θ = 0. On a cell C such that λ > 0 and θ = 0, we have that

f(x, y) = yλa(x)u(x, y) and |u(x, y) − 1| < min(α−2, yµ|b(x)|)

+ + for all (x, y) ∈ C. Since λ ∈ Q , we may write λ = p/q, for some p, q ∈ Z . Put H := {hq : h ∈ H} and notice that f(x, y) f(x, y)q ∈ H if and only if ∈ H0. yλ yp By basic abstract algebra, we have that H = H0 ∪ αλH0 ∪ ... ∪ α(q−1)λH0, so f(x, y)q f(x, y)q ∈ H0 if and only if ∈ αkλH yp yp f(x, y)q for some k ∈ {0, 1, . . . , q − 1}. Since yp ∈ H, we have that ∈ αkλH is yp f(x, y)q equivalent to ∈ H. This reduces to Case 1.1. αkλ

Case 1.3: λ > 0, θ 6= 0. On a cell C such that λ > 0 and θ 6= 0, we have without loss of generality that

f(x, y) = (θ(x) − y)λa(x)u(x, y), where 43 0 < θ(x) − y < θ(x)/α2 and |u(x, y) − 1| < min(α−2, |y − θ(x)|µ|b(x)|)

By the inductive hypothesis, the set {(x, y) ∈ C ∩ Hm−1 : 0 < θ(x) − y < θ(x)/α2} is definable in the structure (H, · , <), and is in fact the graph of a function definable in the structure (H, · , <). That the graph of f intersected with Hm is definable in

(H, · , <) follows immediately.

Case 2: Thin cells. We now suppose that 0 is a limit point of A and that A ∩ Hm is the set

{(x, y, z) ∈ Hm−2 × H × H :(x, y) ∈ C and z is the closest element of H to f(x, y)},

m−1 where C is a cell in (0, 1) definable in R and the function f : C → R is definable in R. By the inductive hypothesis, we have that C∩Hm−1 is definable in (H, · , <). Again by van den Dries and Speissegger [12], we may assume without loss of generality that

f(x, y) = |y − θ(x)|λa(x)u(x, y), and

|u(x, y) − 1| < min(α−2, |y − θ(x)|µ|b(x)|) for all (x, y) ∈ C.

Case 2.1: λ = 0. In this case we have (x, y, z) ∈ A ∩ Hm if and only if (x, y) ∈

C ∩ Hm−1 and z is the closest element of H to a(x)u(x, y). Ranging over all y such that (x, y) ∈ C, there are at most two possible values for z, the closest element of

H to f(x, y): either it is the closest element of H to a(x), or it is the second closest element of H to a(x). Of these two values, one is larger than a(x) and one is smaller. 44 m−1 Since the set {(x, y) ∈ C : a(x) ≤ f(x, y)} is a deﬁnable subset of R , we may partition C into two subsets, one on which a ≤ f and one on which a > f. On the former, we always take z to be the larger of the two options, and on the latter we take z to be the smaller of the two options. Thus A∩Hm is deﬁnable in the structure

(H, · , <).

Case 2.2: λ > 0. On a cell C such that λ > 0 we have that

f(x, y) = yλa(x)u(x, y) and |u(x, y) − 1| < min(α−2, |y − θ(x)|µ|b(x)|) for all (x, y) ∈ C. One handles this case by treating the function (x, y) 7→ f(x, y)/yλ as in Case 2.1.

Case 3: Other cells. Suppose now that there is a cell C and a deﬁnable function f : C → R such that f > 0 and

m−1 A = {(x, z) ∈ R × R : x ∈ C and f(x) < z}.

We handle this case by approximating the lower boundary of A by a function g :

C ∩ Hm−1 → H, deﬁnable in the structure (H, · , <), whose graph is contained within a thin cell (as in Case 2) above the graph of f. We deﬁne this function thus: for

(x, z) ∈ A ∩ Hm, let g(x) be the smallest element of H greater than f(x). Then the graph of g is Hm intersected with the cell

m−1 {(x, z) ∈ R × R : x ∈ C and f(x) < z ≤ αf(x)}, 1 + α and for each x ∈ C we have that g(x) is the closest element to f(x) such that 2 f(x) 6= g(x) By Cases 1 and 2, the graph of g is thus deﬁnable in the structure

(H, · , <). Since

m m−1 A ∩ H = {(x, z) ∈ R × R : x ∈ C and g(x) < z}, 45 we have that A ∩ Hm is deﬁnable in (H, · , <) as well.

The case where there is a cell C and a deﬁnable function f : C → R such that f > 0 and

m−1 A = {(x, z) ∈ R × R : x ∈ C and f(x) > z} is handled similarly, and the case where there is a cell C and deﬁnable functions f : C → R such that 0 < f < g and

m−1 A = {(x, z) ∈ R × R : x ∈ C and g(x) > z > f(x)} is handled by taking intersections. As any cell is either of one of these forms or is a graph of a function over a lower dimensional cell, we have shown that A ∩ Hm is deﬁnable in (H, · , <).

By induction, this is true for any m ∈ N. Thus, the structure induced on H in (R,H) is interdeﬁnable with (H, · , <) as claimed.

As a corollary, we can now strengthen the conclusion of Theorem 4.1.10:

n Theorem 4.2.1. Let A ⊆ R be a set deﬁnable in the structure (R,H). Then there

mi are ﬁnitely many cells Ci ⊆ (0, 1) and ﬁnitely many continuous functions (fi :

di n mi di Ci ×(0, 1) → R )i∈I such that A is the disjoint union of the images fi(H ×(0, 1) ) and such that for each i ∈ I

• The function fi is deﬁnable in the structure R.

p • For each y ∈ Ci, the function x → fi(y, x) is injective and C .

m di • The function fi is injective on the set (C ∩ H ) × (0, 1) .

m d n m d Proof. Fix a function f : (0, 1) × (0, 1) → R . For each y ∈ f(H × R ), consider the set

m d Hy := {h ∈ H : ∃x ∈ (0, 1) such that y = f(h, x)}. 46 By Miller [25], the structure (R,H) admits deﬁnable choice. Thus there is a function

m d m g : f(H × R ) → H deﬁnable in the structure (R,H) such that g(y) ∈ Hy for m d all y ∈ f(H × R ). Let S be the image of the map G :(h, x) 7→ (g(f(h, x)), x). As in Corollary 4.1.7, we need only consider the case where the function g is given piecewise by tuples of terms in the language of (R, Λ).

Let t(y) be one such term. By Lemma 4.1.5, we have that there is a k ∈ N and m 0 k there are R-terms s, s1, . . . , sk such that for each y ∈ R there is an x ∈ H such 0 that t(y) = s(y, x ), and si(y, x1, . . . , xi−1) < xi ≤ αsi(y, x1, . . . , xi−1) for i = 1, . . . , k. One sees that for each y there is at most one x0 ∈ Hk satisfying these requirements.

Thus we may replace the term t(y) by the L-term s(y, x0) without altering the value of the function G. We continue in this way to replace each LΛ-term t(y) with a L-term s(y, x0) with appropriate requirements on x0. This results in a function G¯ :

0 m d (h, x , x) 7→ (g(f(h, x)), x) such that for each h ∈ H and each x ∈ R there is a 0 k 0 unique x ∈ R such that (h, x , x) is in the domain of G¯. Thus the image of G is ¯ given by the function G deﬁnable in R. By the deﬁnition of S, the restriction fS is injective and it has the same image as f.

4.3 Deﬁnable estimates for d-volume

n For a set A ⊆ R , we define its d-dimensional volume, vold(A), to be its d-dimensional Hausdorff measure with the usual scaling factor. Let f : (0, 1)m+d → (0, 1)n be a function as in the conclusion of Theorem 4.2.1. Our goal is to determine when the d-dimensional volume of the set f(Hm × (0, 1)d) is finite. We will utilize a powerful theorem in o-minimality to reduce to the case where f satisfies a uniform Lipschitz condition.

By o-minimality, we decompose the domain of f into cells in such a way that the image of each function fh(x) := f(h, x) projects onto each d-dimensional coordinate 47 n plane Pi in R either injectively or in a set with empty interior. Fix such a decomposition C and let cell C ∈ C. Let πi denote projection on coordinate plane Pi. Then by the Cauchy-Binet theorem we have that

n max (vol π (f(C))) ≤ vol f(C) ≤ max (vol π (f(C))), i d i d d i d i so the d-dimensional volume of f(C) is finite if and only if the d-dimensional volume of each of its projections πi(f(C)) is finite. As our only interest is determining whether or not vold f(C) is finite, we may without loss of generality assume that d = n.

l Recall that a diﬀerentiable function φ : R → R is uniformly Lipschitz with l constant M if for all x ∈ R and for j ∈ {1, . . . , l} we have that

∂φ (x) ≤ M. ∂xj

k A cell S ⊆ R is called a regular M-cell cell if each of the functions deﬁning the boundaries of the cell is both of class C1 and is uniformly Lipschitz with constant M.

Theorem 4.3.1 (Paw lucki [26]). Let k ∈ N. Then there is number Mk such that for k any set X ⊆ R there is a deﬁnable partitioning X = S1 t ... t Sk t Σ such that Σ has no interior and each Si is, after a permutation of coordinates, an open regular

Mk-cell.

We will show that f may be assumed to be uniformly Lipschitz. Let g : C →

(0, 1)m+n given by g(h, x) = (h, f(h, x)). After a Paw lucki decomposition, we may assume that S := g(C) either has no interior or that it is, after a permutation of coordinates, an open regular Mm+n-cell. In the former case, we may assume that S is a cell of dimension strictly less than m + n. If the deficiency in dimension comes from one of the first m variables, then Theorem 4.1.2 allows us to reduce to a lower- dimensional case by projecting in the appropriate variable. If the deficiency occurs in the last n variables, then the d-dimensional volume of S ∩ (Hm × (0, 1)n) is 0, and 48 so S can be ignored for volume calculations. Thus we may assume that S is, after a permutation of coordinates, an open regular Mm+n-cell.

Now we will give a parametrization of S. Let η1, . . . , ηm+n be the permuted

i−1 variables in which S is a regular Mm+n-cell. For i = 1, . . . , m+n, let φi, ψi : (0, 1) →

(0, 1) be the cell boundary functions for S. That is, (η1, . . . , ηn+m) ∈ S if and only if

φ1 < η1 < ψ1

φ2(η1) < η2 < ψ2(η1) ......

φm+n(η1, . . . , ηn+m−1) < ηm+n < ψm+n(η1, . . . , ηn+m−1)

n+m We deﬁne the function F : (0, 1) → S as follows, writing F = (F1,...,Fn+m).

If ηk+1 came from one of the ﬁrst m coordinates under the permutation mentioned above, then deﬁne   νk+1, if φk+1(ν1, . . . , νk) < νk+1 < ψk+1(ν1, . . . , νk) Fk+1(ν1, . . . , νm+n) :=  0, otherwise

If ηk+1 instead came from one of the last n coordinates under the permutation mentioned above, then set

Fk+1(ν1, . . . , νm+n) := (1 − νk)φk+1 + νkψk+1

Deﬁne G : (0, 1)m+n → (0, 1)m+n to be F taken in the variables (h, x) from before the permutation mentioned above was used, with the additional condition that

G(h, x) is not defined if any of the coordinate functions Gi(h, x) = 0, i = 1, . . . , m. The function G(h, x) is then uniformly Lipschitz on its bounded support, and so it has a uniformly continuous extension to the boundary of its support, which we will ¯ m henceforth denote as G. For each h ∈ (0, 1) , define the function Gh(x) := π(G(h, x)), where π is projection on the last n coordinates. One easily sees that the Jacobian 49 | det DGh| is the product of uniformly Lipschitz functions |ψk − φk|, and so is itself uniformly Lipschitz. Let D be the projection of the support of G on the first m coordinates. By elementary Calculus, we thus have that the function V : D → R defined by Z V (h) := | det DGh| (0,1)n n is uniformly Lipschitz, and V (h) = voln(G({h} × (0, 1) ). As a consequence, we have that V has a continuous extension to the boundary of its support.

The function V may not be deﬁnable, so we will instead use this Lipschitz continuity to estimate V by functions that are deﬁnable:

+ Lemma 4.3.2. Let the function V : D → R be as above. Then there are functions

L, U : D → R, both deﬁnable in the structure R, such that

• L ≤ V ≤ U

• For all z ∈ cl D, lim U(h) = 0 if lim V (h) = 0 h→z h→z

• For all h ∈ D, 0 < L(h) if 0 < V (h)

m Proof. Deﬁne the function U : D → R by U(h) = max{| det DGh| : x ∈ (0, 1) }. Clearly U is deﬁnable in the structure R and U ≥ V . By the Lipschitz continuity of

V and | det DGh| we have that for all z ∈ cl D, lim U(h) = 0 if lim V (h) = 0. h→z h→z Now deﬁne the function L : D → R by setting L(h) to be the least upper bound of the set of volumes of all boxes that will ﬁt inside the set G((0, 1)m). Clearly L is

m deﬁnable in the structure R and L ≤ V . Since Gh((0, 1) ) is an open set for all h, we have that for all h ∈ D, 0 < L(h) if 0 < V (h).

50 4.4 Dimension and volume

Let G, G¯, and V be as in the previous section. For each z ∈ fr Hm, we deﬁne the set

m m n Lz(G) := {y ∈ R : ∀ε > 0, ∃h ∈ D ∩ H , ∃x ∈ (0, 1) |G(h, x) − y| < ε}.

m n n+m Since the image G(H × (0, 1) )is an embedded submanifold of R , we have that the frontier of G(Hm × (0, 1)n) is the union

[ d [ fr(G({h} × (0, 1) )) ∪ Lz(G). h∈Hm z∈fr Hm

Since the n-dimensional volume of fr(F¯({h} × (0, 1)n)) is 0, we focus our attention on the set Lz(G). By work in the last section, one sees that the set Lz(G) is just the image of the set fr Hm under the function G¯ that extends G continuously to the boundary of its support. By Lipschitz continuity, we thus have that for each

m z ∈ fr H that voln(Lz(G)) = 0 if and only if lim V (h) = 0. Thus: h→z, h∈(D∩Hm)

Proposition 4.4.1. Let z ∈ fr(D ∩ Hm). Then lim V (h) = 0 if and only if h→z, h∈(D∩Hm) the limit set Lz(G) has zero n-dimensional volume.

Thus, to prove Theorem 4.1.1, it suﬃces by Proposition 4.4.1 and Lemma 4.3.2 to prove the following claim about deﬁnable functions:

Theorem 4.4.2. Let f : (0, 1)m → [0, 1) be deﬁnable in the structure R. Then

X f(h) < ∞ h∈Hm if and only if for all h ∈ fr Hm we have that lim f(z) = 0 z→h,z∈Hm X Proof. If f(h) converges, then by the Vanishing Criterion it must be the case h∈Hm that for all h ∈ fr Hm we have that lim f(z) = 0. Thus we suppose that for z→h,z∈Hm m m all h ∈ fr H we have that lim f(z) = 0. For h = (h1, . . . , hm) ∈ H , let z→h,z∈Hm 51 |h| = min{h1, . . . , hm}. Notice that for each n ∈ N we have that the number of m −n m elements in the set {h ∈ H : |h| = α } is less than n , so for each n ∈ N

X f(h) < nm max{f(h): h ∈ Hm and |h| = α−n}, h∈Hm,|h|=α−n and thus that

X X f(h) < nm max{f(h): h ∈ Hm and |h| = α−n}. h∈Hm α−n∈H

Clearly for all z ∈ (0, 1)m, we have that Λ(f(z)) ≤ f(z) < αΛ(f(z)), where Λ is as deﬁned in Section 4.1. Thus by the Comparison Test is suﬃces to consider the case

m m where f(H ) ⊆ H. By Theorem 4.1.2, in this case the restriction f(supp(f) ∩ H ) is deﬁnable in the structure (H, · , <), and so the function g : H → H deﬁned by

m m g(x) := max{f(supp(f) ∩ H )(h): h ∈ H and |h| = x} is also deﬁnable in the structure (H, · , <). Since f is deﬁnable in (H, · , <) and for all h ∈ fr Hm we have that lim f(z) = 0, we must have that lim g(x) = 0. By z→h,z∈Hm x→0+ Theorem 4.1.3, we may assume that there are rational numbers a and b such that

−n a+bn for all n ∈ N we have that g(α ) = α . Since lim g(x) = 0, we must have that x→0+ b < 0. Notice that

X X nm max{f(h): h ∈ Hm and |h| = α−n} = nmg(α−n), α−n∈H α−n∈H and so ∞ X X f(h) < nmαa+bn, h∈Hm n=0 X which converges by the Ratio Test. Thus the sum f(h) converges. h∈Hm

52 BIBLIOGRAPHY

[1] James Ax, On Schanuel’s conjectures, Ann. of Math. (2) 93 (1971), 252–268. MR0277482 [2] Ricardo Bianconi, Nondefinability results for expansions of the field of real numbers by the exponential function and by the restricted sine function, J. Symbolic Logic 62 (1997), no. 4, 1173–1178. MR1617985 [3] , Undefinability results in o-minimal expansions of the real numbers, Ann. Pure Appl. Logic 134 (2005), no. 1, 43–51. MR2132914 [4] Raf Cluckers, Presburger sets and p-minimal fields, J. Symbolic Logic 68 (2003), no. 1, 153–162. MR1959315 (2003m:03062) [5] Lou van den Dries, Remarks on Tarski’s problem concerning (R, +, ·, exp), Logic colloquium ’82 (Florence, 1982), Stud. Logic Found. Math., vol. 112, North-Holland, Am- sterdam, 1984, pp. 97–121. MR762106 (86g:03052) [6] , The field of reals with a predicate for the powers of two, Manuscripta Math. 54 (1985), no. 1-2, 187–195. MR808687 (87d:03098) [7] , A generalization of the Tarski-Seidenberg theorem, and some nondefinability results, Bull. Amer. Math. Soc. (N.S.) 15 (1986), no. 2, 189–193. MR854552 (88b:03048) [8] , On the elementary theory of restricted elementary functions, J. Symbolic Logic 53 (1988), no. 3, 796–808. MR960999 (89i:03074) [9] , Tame topology and o-minimal structures, London Mathematical Society Lec- ture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998. MR1633348 (99j:03001) [10] , Dense pairs of o-minimal structures, Fund. Math. 157 (1998), no. 1, 61–78. MR1623615 (2000a:03058) [11] Lou van den Dries and Ayhan Günaydın, The fields of real and complex numbers with a small multiplicative group, Proc. London Math. Soc. (3) 93 (2006), no. 1, 43–81. MR2235481 (2007i:03039) [12] Lou van den Dries and Patrick Speissegger, O-minimal preparation theorems, Model theory and applications, Quad. Mat., vol. 11, Aracne, Rome, 2002, pp. 87–116. MR2159715 (2006m:03063) [13] Tom Foster, Uniform model-completeness for the real field expanded by power functions, J. Symbolic Logic 75 (2010), no. 4, 1441–1461. MR2767978 (2012f:03057) 53 [14] Ayhan Günaydın, Model theory of fields with multiplicative groups, Ph.D. thesis, Uni- versity of Illinois at Urbana-Champaign, 2008. [15] Jacques Herbrand, Recherches sur la théoriede la démonstration, Trav. Soc. Sci. Lett. Varsovie Cl. III 33 (1930), 1–128. [16] Philipp Hieronymi, Defining the set of integers in expansions of the real field by a closed discrete set, Proc. Amer. Math. Soc. 138 (2010), no. 6, 2163–2168. MR2596055 [17] Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR1321597 (96e:03057) [18] David Marker, Model theory, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002. An introduction. MR1924282 (2003e:03060) [19] Chris Miller, Expansions of the real field with power functions, Ann. Pure Appl. Logic 68 (1994), no. 1, 79–94. MR1278550 [20] , A growth dichotomy for o-minimal expansions of ordered fields, Logic: from foundations to applications (Staffordshire, 1993), Oxford Sci. Publ., Oxford Univ. Press, New York, 1996, pp. 385–399. MR1428013 (98a:03052) [21] , Tameness in expansions of the real field, Logic Colloquium ’01, Lect. Notes Log., vol. 20, Assoc. Symbol. Logic, Urbana, IL, 2005, pp. 281–316. MR2143901 [22] , Avoiding the projective hierarchy in expansions of the real field by sequences, Proc. Amer. Math. Soc. 134 (2006), no. 5, 1483–1493 (electronic). MR2199196 (2007h:03065) [23] , Expansions of o-minimal structures on the real field by trajectories of linear vector fields, Proc. Amer. Math. Soc. 139 (2011), no. 1, 319–330. [24] , Basics of o-minimality and Hardy fields, Lecture notes on O-minimal structures and real analytic geometry, Fields Inst. Commun., vol. 62, Springer, New York, 2012, pp. 43–69. MR2976990 [25] , Definable choice in d-minimal expansions of ordered groups, preliminary re- port, available at http://www.math.ohio-state.edu/~miller. [26] Wieslaw Pawlucki, Lipschitz cell decomposition in o-minimal structures. I, Illinois J. Math. 52 (2008), no. 3, 1045–1063. MR2546024 (2010j:14099) [27] Anand Pillay and Charles Steinhorn, Definable sets in ordered structures, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 1, 159–162. MR741730 (86c:03033) [28] Julia Robinson, The undecidability of algebraic rings and fields, Proc. Amer. Math. Soc. 10 (1959), 950–957. MR0112842 (22 #3691) [29] Raphael M. Robinson, The undecidability of pure transcendental extensions of real fields, Z. Math. Logik Grundlagen Math. 10 (1964), 275–282. MR0172803 (30 #3021) [30] Patrick Speissegger, The Pfaffian closure of an o-minimal structure, J. Reine Angew. Math. 508 (1999), 189–211. MR1676876 (2000j:14093) [31] Alfred Tarski, A decision method for elementary algebra and geometry, University of California Press, Berkeley and Los Angeles, Calif., 1951. 2nd ed. MR0044472 (13,423a) 54 [32] Michael A. Tychonievich, Defining additive subgroups of the reals from convex subsets, Proc. Amer. Math. Soc. 137 (2009), no. 10, 3473–3476. MR2515416 (2010m:03076) [33] , The set of restricted complex exponents for expansions of the reals, Notre Dame J. Formal Logic 53 (2012), no. 2, 175-186. [34] A. J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc. 9 (1996), no. 4, 1051–1094. MR1398816