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Jarkko Peltomäki Privileged Words and Sturmian Words Jarkko Peltomäki Privileged Words and Sturmian Words Turku Centre for Computer Science TUCS Dissertations No 214, August 2016 Privileged Words and Sturmian Words Anadromit ja Sturmin sanat Jarkko Peltomäki To be presented, with the permission of the Faculty of Mathematics and Natural Sciences of the University of Turku, for public criticism in Tauno Nurmela Hall (Lecture Hall I) on August 19th, 2016, at 12 noon. Turun yliopisto Matematiikan ja tilastotieteen laitos FI-20014 Turku, Finland 2016 Supervisors Professor Tero Harju Matematiikan ja tilastotieteen laitos Turun yliopisto FI-20014 Turku Finland Professor Luca Zamboni Institut Camille Jordan Université Claude Bernard Lyon 1 F-69622 Villeurbanne Cedex France Reviewers Professor Valérie Berthé CNRS – IRIF Université Paris 7- Paris Diderot Case 7014, F-75205 Paris Cedex 13 France Associate Professor Narad Rampersad Department of Mathematics and Statistics University of Winnipeg Winnipeg, MB, R3B 2E9 Canada Opponent Professor Dirk Nowotka Institut für Informatik Christian-Albrechts-Universität zu Kiel DE-24098 Kiel Germany The originality of this thesis has been checked in accordance with the University of Turku quality assurance system using the Turnitin OriginalityCheck service. Painosalama Oy, Turku ISBN 978-952-12-3421-7 (PRINT) ISBN 978-952-12-3422-4 (ELECTRONIC) ISSN 1239-1883 Abstract This dissertation has two almost unrelated themes: privileged words and Stur- mian words. Privileged words are a new class of words introduced recently. A word is privileged if it is a complete first return to a shorter privileged word, the shortest privileged words being letters and the empty word. Here we give and prove almost all results on privileged words known to date. On the other hand, the study of Sturmian words is a well-established topic in combinatorics on words. In this dissertation, we focus on questions concerning repetitions in Sturmian words, reproving old results and giving new ones, and on establishing completely new research directions. The study of privileged words presented in this dissertation aims to derive their basic properties and to answer basic questions regarding them. We explore a connection between privileged words and palindromes and seek out answers to questions on context-freeness, computability, and enumeration. It turns out that the language of privileged words is not context-free, but privileged words are recognizable by a linear-time algorithm. A lower bound on the number of binary privileged words of given length is proven. The main interest, however, lies in the privileged complexity functions of the Thue-Morse word and Sturmian words. We derive recurrences for computing the privileged complexity function of the Thue-Morse word, and we prove that Sturmian words are characterized by their privileged complexity function. As a slightly separate topic, we give an overview of a certain method of automated theorem-proving and show how it can be applied to study privileged factors of automatic words. The second part of this dissertation is devoted to Sturmian words. We exten- sively exploit the interpretation of Sturmian words as irrational rotation words. The essential tools are continued fractions and elementary, but powerful, results of Diophantine approximation theory. With these tools at our disposal, we re- prove old results on powers occurring in Sturmian words with emphasis on the fractional index of a Sturmian word. Further, we consider abelian powers and abelian repetitions and characterize the maximum exponents of abelian powers with given period occurring in a Sturmian word in terms of the continued frac- tion expansion of its slope. We define the notion of abelian critical exponent for Sturmian words and explore its connection to the Lagrange spectrum of irrational numbers. The results obtained are often specialized for the Fibonacci word; for instance, we show that the minimum abelian period of a factor of the Fibonacci word is a Fibonacci number. In addition, we propose a completely new research topic: the square root map. We prove that the square root map preserves the lan- guage of any Sturmian word. Moreover, we construct a family of non-Sturmian optimal squareful words whose language the square root map also preserves. This construction yields examples of aperiodic infinite words whose square roots are periodic. i ii Tiivistelmä Tässä väitöskirjassa on kaksi lähestulkoon erillistä pääteemaa: anadromit ja Stur- min sanat. Anadromi on sana, joka on täydellinen paluu lyhyempään anadromiin – lyhimmät anadromit muodostuvat kirjaimista ja tyhjästä sanasta. Tämä väitös- kirja sisältää todistukset lähes kaikille näitä aivan vastikään määriteltyjä sanoja koskeville tuloksille. Sturmin sanat taasen ovat vakiintuneempi tutkimuskohde sanojen kombinatoriikan alalla. Sturmin sanoja koskien tässä väitöskirjassa esite- tään uusia todistuksia tunnetuille tuloksille – kuten toistojen luonnehdinnalle – sekä todistetaan aivan uusia tuloksia ja avataan uusia tutkimussuuntauksia. Anadromeja koskevassa osuudessa johdetaan näiden sanojen perusominai- suudet ja tutkitaan aihetta koskevia perustavia kysymyksiä. Keskiöissä ovat ana- dromien ja palindromien yhteydet sekä kysymykset liittyen kontekstittomuu- teen, laskettavuuteen ja anadromien lukumäärään eräissä kielissä. Osoittautuu, että anadromien kieli ei ole kontekstiton, mutta anadromin voi tunnistaa line- aariaikaisella algoritmilla. Lisäksi johdetaan alaraja tiettyä pituutta olevien bi- näärianadromien lukumäärälle. Keskeisintä on kuitenkin Thuen–Morsen sanan ja Sturmin sanojen anadromikompleksisuusfunktioiden tutkimus. Tässä työssä johdetaan ryhmä rekursioyhtälöitä Thuen–Morsen sanan anadromikompleksi- suuden laskemiseksi ja luonnehditaan Sturmin sanat niiden anadromikomple- ksisuusfunktion avulla. Lopuksi esitellään erästä verrattain uutta automaattista teoreemantodistusmenetelmää ja sovelletaan sitä automaattisten sanojen anadro- mitekijöiden tutkimiseksi. Väitöskirjan jälkimmäisessä osassa käsitellään Sturmin sanoja. Sturmin sanat käsitetään erityisesti irrationaalisina rotataatiosanoina, mikä tarkoittaa, että ket- jumurtoluvut ja niihin liittyvä teoria rationaaliapproksimaatioista ovat sovellet- tavissa Sturmin sanoihin. Näitä lukuteorian menetelmiä runsaasti hyödyntäen tässä väitöskirjassa sekä johdetaan aivan uusia tuloksia Sturmin sanoista että esitetään paranneltuja todistuksia tunnetuille lauseille. Ensimmäisenä teemana ovat toistot ja abelin toistot. Uusia menetelmiä käyttäen Sturmin sanojen toisto- jen ja rationaali-indeksin luonnehdinnoille esitetään uudet todistukset. Seuraa- vaksi uutena tuloksena määritetään Sturmin sanassa annettua jaksoa vastaavat mahdolliset abelin toistojen eksponentit tämän sanan kaltevuuden ketjumurtolu- kukehitelmän perusteella. Lisäksi osoitetaan Sturmin sanojen ns. kriittisen abe- lin eksponentin yhteys irrationaalilukujen Lagrangen spektriin. Monia tuloksia tarkastellaan Fibonaccin sanan erityistapauksessa. Osoittautuu esimerkiksi, et- tä Fibonaccin sanan tekijän lyhin abelin jakso on aina Fibonaccin luku. Lopuksi määritellään täysin uusi käsite: Sturmin sanojen neliöjuurifunktio. Tämä erikoi- nen funktio yllättäen säilyttää Sturmin sanojen kielen. Yleisemmässä tapauksessa konstruoidaan perhe optimaalisia ja neliöllisiä sanoja, joiden kielen neliöjuuri- funktio säilyttää mutta jotka eivät ole Sturmin sanoja. Tämä konstruktio antaa esimerkin jaksottomasta äärettömästä sanasta, jonka neliöjuuri on jaksollinen. iii iv Acknowledgments This dissertation has been done under the guidance and supervision of Professors Tero Harju and Luca Zamboni (while he had a position in Turku). I thank you the most for freedom to work independently. Tero I thank especially for supervis- ing my master’s thesis, which served as a good introduction to combinatorics on words, and for his, often very calligraphic, lectures with all those dry jokes. I thank Luca for his uncanny insight in proposing good research problems. Professor Juhani Karhumäki is acknowledged for the excellent work condi- tions at the math department and for his active role in TUCS and the doctoral programme MATTI; I gratefully acknowledge the splendid support from Juhani’s projects. The current head of the department, Professor Iiro Honkala, receives my warm thanks for continuing to support the department’s good atmosphere. I also thank all researchers involved with FUNDIM for the great community we have. In particular, I want to thank Professor Jarkko Kari for introducing me to cellular automata, tilings, and automata theory. In addition, I am grateful for the financial support of TUCS, University of Turku Graduate School (UTUGS), Department of Mathematics and Statistics, and Suomen kulttuurirahasto. I thank my coauthors, particularly Professor Jeffrey Shallit, Associate Profes- sor Gabriele Fici, and fellow Ph.D. student Markus Whiteland, for the obvious reasons. I also thank Doctors Svetlana Puzynina and Alessandro De Luca and all the other kind researchers I have met at various conferences or otherwise. I am grateful for Professor Valérie Berthé and Associate Professor Narad Rampersad for the time spent on reviewing this dissertation. I thank Professor Dirk Nowotka for agreeing to act as my opponent. Obviously, I am indebted to the (ex-)administrative staff of our department, Tuire Huuskonen, Sonja Vanto, Lasse Forss, and Laura Kullas. Special thanks to Doctor Arto Lepistö for our conversations over mathematics, computer hard- ware, the GNU/Linux operating system, etc. By mentioning the GNU/Linux system, it is
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