Using Automata Theory to Solve Problems in Additive Number Theory by Aayush Rajasekaran A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics in Computer Science Waterloo, Ontario, Canada, 2018 c Aayush Rajasekaran 2018 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract Additive number theory is the study of the additive properties of integers. Perhaps the best-known theorem is Lagrange's result that every natural number is the sum of four squares. We study numbers whose base-k representations have certain interesting proper- ties. In particular, we look at palindromes, which are numbers whose base-k representations read the same forward and backward, and binary squares, which are numbers whose binary representation is some block repeated twice (like (36)2 = 100100). We show that all natural numbers are the sum of four binary palindromes. We also show that all natural numbers are the sum of three base-3 palindromes, and are also the sum of three base-4 palindromes. We also show that every sufficiently large natural number is the sum of four binary squares. We establish these results using virtually no number theory at all. Instead, we construct automated proofs using automata. The general proof technique is to build an appropriate machine, and then run decision algorithms to establish our theorems. iii Acknowledgements If the statement of authorship that opens this thesis gives the suggestion that this was a \one-man effort”, let me hastily dispel that notion. There are many individuals and institutions that I have left unthanked for too long, and it is only proper that I attempt to right that wrong. First and foremost, I would like to thank my supervisor, Dr. Jeffrey Shallit, for every- thing he has given me over the past 2 years. It is no exaggeration to say that this thesis would not have been possible without you. Your breadth of knowledge, passion for re- search, and incredible assiduity are all truly inspiring. I will always remember the day you sent me an email with the subject \cool problem" that included attachments describing nested-word automata. The past 2 years under your supervision have seen many weekly meetings, a few published papers, and nearly a dozen trivia nights. I will keep these mem- ories with me, and will remain grateful to you for all the opportunities, experiences, and skills you have given me. Thank you, Jeff. The work described in this thesis was performed in collaboration with some remarkable people. My co-author, Tim Smith, came up with many of the ideas that were critical in proving our results. I'm also grateful to the other mathematicians I got to collabo- rate with during my Master's study: Narad Rampersad, Dirk Nowotka, and Madhusudan Parthasarathy. I would like to think Dr. Eric Blais and Dr. Jason Bell for agreeing to read this the- sis. More generally, I would like to think the faculty and staff at the Cheriton School of Computer Science. I would also like to acknowledge the excellent psychological support I received during my course of therapy at the Centre for Mental Health Research. The University of Waterloo has been my home for 7 years now, and leaving is far more bitter than sweet. There are two individuals whose friendships I savagely exploited for assistance at every stage of my graduate studies. Thank you, Ming-Ho Yee, for having been a big brother of sorts since 2011. How do most graduate students survive without your advice? I'm glad I never had to find out. Rein Otsason, you are in every page of this thesis. The story of my Master's study is the story of our relationship. Thank you. And finally, I would like to thank the many people who make my daily life the joy that it is. Perhaps the most valuable service one human being can provide to another is to make them laugh. The many, many people I ungratefully demand that service from include Arjun, Ian, Christine, Shelby, Nick P., Nick S., Anubhav, and most recently, Jacky Jack. Thank you all. iv For the constants. Moo. Dada and Anna. Kira and Faith. v Table of Contents List of Figures ix 1 Introduction1 1.1 Contributions of the Thesis..........................1 1.2 Definitions and Notation............................1 1.3 Overview of Additive Number Theory.....................2 1.4 Palindromes...................................3 1.5 Squares.....................................4 1.6 Our Approach..................................4 2 Automata Theory5 2.1 Finite Automata................................5 2.2 Nested Word Automata............................6 3 Results on Binary Palindromes9 3.1 The Idea.....................................9 3.2 The Result.................................... 10 3.3 The Machines.................................. 12 3.4 An Example Input............................... 14 3.5 Checking Correctness.............................. 17 vi 4 Palindromes in Larger Bases 19 4.1 The Folding Trick................................ 19 4.2 The Results................................... 20 4.3 The Machines.................................. 22 4.4 Example Inputs................................. 24 5 Results on Squares 28 5.1 Introduction................................... 28 5.2 Proof of Lemma 12............................... 29 5.2.1 Odd-length Inputs........................... 30 5.2.2 Even-length Inputs........................... 33 5.3 Checking Correctness.............................. 35 5.4 Other Results on Squares............................ 36 6 Some Results from Walnut 39 6.1 Automatic Sequences and Walnut....................... 39 6.2 The Approach.................................. 40 6.3 Results on the Fibonacci word......................... 42 6.4 Evil and Odious Numbers........................... 43 7 Discussion 46 7.1 Limitations of this Proof Technique...................... 46 7.1.1 Time and Space Complexity...................... 46 7.1.2 Limitations of the Machines...................... 47 7.2 Future Work on Palindromes.......................... 47 7.3 Future Work on Squares............................ 47 References 49 APPENDICES 53 vii A Walnut automata files 54 A.1 Fibonacci Numbers............................... 54 A.2 Numbers of the Form 4n − 1.......................... 54 A.3 Evening Numbers................................ 55 viii List of Figures 2.1 An illustration of M ..............................6 2.2 A nested-word automaton for the language f0n12n : n ≥ 1g ........7 6.1 The automaton outputting the Thue-Morse word.............. 40 6.2 The automaton outputting the Rudin-Shapiro word............. 41 6.3 The result automaton.............................. 41 ix Chapter 1 Introduction 1.1 Contributions of the Thesis The primary results presented in this thesis relate to expressing numbers as sums of bi- nary palindromes and binary squares (defined in Sections 1.4 and 1.5). In Theorem1, we show that all natural numbers are the sum of at most 4 binary palindromes. Theo- rem4 establishes a similar result for binary squares. In Chapter6, we use the automatic theorem-proving software Walnut[30] to establish a variety of results regarding the additive properties of automatic sequences. The results regarding palindromes first appeared in the author's paper with Shallit and Smith [35]. The results regarding squares first appeared in the author's paper with Madhusudan, Nowotka, and Shallit [27]. Some sections of this thesis have been copied verbatim from these papers. 1.2 Definitions and Notation We present some standard definitions in this section. An alphabet, usually denoted Σ, is a finite, non-empty set of symbols. We denote the alphabet of size k as Σk, with the letters implicitly assumed to be 0; 1 : : : k − 1. We denote the set of all possible (finite) words on an alphabet Σ as Σ∗. For integers n; k ≥ 0, we denote the canonical base-k representation of n as (n)k. The canonical base-k representation does not include any leading zeros, and starts with the 1 most significant digit. For instance, (39)2 = 100111. The canonical representation of 0 in any base is the empty string, which we denote . Note that (n)k is a word on the alphabet Σk. 1.3 Overview of Additive Number Theory One can easily see that every natural number is the sum of at most 2 odd integers. It is also self-evident that a similar statement cannot be made about the even integers. There are infinitely many natural numbers that cannot be written as the sum of any number of even integers. A non-trivial statement along these lines is Lagrange's famous theorem that every natural number is the sum of four squares [16, 29]. As an example, the natural number 983 = 212 + 222 + 72 + 32. These are the kinds of results studied under the field of additive number theory. Formally, additive number theory is the study of the additive properties of integers [31]. For a given T ⊆ N, an additive basis of order h is a subset S ⊆ N such that every n 2 T is the sum of h members, not necessarily distinct, of S. The principal problem of additive number theory is as follows: given two subsets S and T , does S form an additive basis of some order h for T , and if so, what is the smallest value of h? It is trivially evident that the natural numbers, N, form an additive basis of order 1 for N. Goldbach's famous conjecture [44] can be phrased in this language. If S is the set of even integers greater than 2, and T is the set of prime numbers, the conjecture asks whether T forms an additive basis for S of order 2. There has been much research in this area, and deep techniques, such as the Hardy-Littlewood circle method [42,4], have been developed to solve problems. We can rephrase Lagrange's theorem in our terminology. The theorem tells us that S = f02; 12; 22; 32;:::g forms an additive basis for N of order 4.