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 By 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 struc- ture 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 sub- manifold 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 2 F, a set of relation symbols R along with positive integers nR for each R 2 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 f(; ); :; _; ^; 9; 8g, 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 2 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 fP(M ): n 2 Ng, 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 f+; −; ·; 0; eg, 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 fvngn2N 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 are L-formulas, then so are (:φ), (φ ^ ), and (φ _ ). • If φ is an L-formula, then for all i 2 N so are (9viφ) and (8viφ). For formulas and , we abbreviate (:φ) _ as ! φ and ( ! φ) ^ (φ ! ) as $ 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 9v or 8v, 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 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) 2 M , we write (a) to indicate that for each k 2 f1; : : : ng, each free occurrence of vik in is replaced by ak (more precisely, new symbols for a1; : : : ; an must be added to the language first).