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Digital Lines, Sturmian Words, and Continued Fractions

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Digital Lines, Sturmian Words, and Continued Fractions UPPSALA DISSERTATIONS IN MATHEMATICS 65 Digital lines, Sturmian words, and continued fractions Hanna Uscka-Wehlou Department of Mathematics Uppsala University UPPSALA 2009 ! "#$%&$' ()0&)01''203$( '4'5$ 67'688918@17A &0A"A%&&BC&#$D "!"E0&B "FG&!'HB6889B0'5!$D'"!A"BI$F vvvr %$ $( Vhyh DT7I 9QR9178P68988B S & & D " $ 0 A$ 0 0$ "$ "DBG"00&'T!'!$"&" DD& < "8BGA$!"!A" WXY %S0&" A&XA0BGAA"& `$D&"&A!"A &!&"&""!"B S%SSD!&`$0"A!!0 0BG&D&&XA&"&"$A$ ! &!&"&"!"!A&BC&#aA!"!" &!& "A"&0&#A&XD&"& $ && `B S%SSSD&!A!$"&"D"$!X A$!D&DDFD$&'AD&"&DA$! b&SSSc! IF'D&&"FD&T!"$&B! &"$A$XT!1W1d%Y!$& "!&D&"&A"&!&"&"!"!A!$"&"D 00B S%SeDADT!"&A0D& < fA&$0!"&D&&$!0& `'&"0!&""0&!"!"$ A0&!BG&T!""D&"$$ $#$$BS%eDF&F&T!"A !"!"'&$A&"#'D&"&$F&!$0B S%eSDA' 0D&&DFD BGA$!AAA#&$A 5!$DBG&D&"&T!""!& !0& `WA%Y&##"A#A&! "!""0B 00$'0'&"&A!'"$ "D'5!$ D'!$"&"D'"&""D''"!A"'g! $'A# hH"FG&!6889 S55i1p8168p9 S5ci9QR9178P68988 WY To "the kids": Milena, Julian, Andrzej, Danielle and Charline, and to my sisters: Joanna, the first of us to do her Ph.D., and Maria, who is still working on hers. Good luck! Cover image: Eight digital straight line segments illustrating the equiv- alence relation based on the run length on all digitization levels (as de- fined in Paper IV), restricted to the first four levels. The eight possible forms of S4 with the length specification (1, 2, 2, 3) are presented. The 0’s and 1’s on the leftmost digital straight line segment give an understand- ing of the relationship between digital lines and the corresponding upper mechanical words (chain codes). List of Papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I Uscka-Wehlou, Hanna, 2007. Digital lines with irrational slopes. Theoretical Computer Science 377, pp. 157–169. II Uscka-Wehlou, Hanna, 2009. Run-hierarchical structure of digital lines with irrational slopes in terms of continued fractions and the Gauss map. Pattern Recognition 42, pp. 2247–2254. III Uscka-Wehlou, Hanna, 2008. A Run-hierarchical Description of Upper Mechanical Words with Irrational Slopes Using Continued Fractions; 15 pp. In Proceedings of 12th Mons Theoretical Computer Science Days (Mons, Belgium), 27–30 August 2008. http://www.jmit.ulg.ac.be/jm2008/index-en.html. IV Uscka-Wehlou, Hanna, 2009. Two equivalence relations on digital lines with irrational slopes. A continued fraction approach to up- per mechanical words. Theoretical Computer Science 410 (38–40), pp. 3655–3669. V Uscka-Wehlou, Hanna, 2009. Continued fractions, Fibonacci num- bers, and some classes of irrational numbers; 14 pp. Manuscript submitted to a journal. VI Uscka-Wehlou, Hanna, 2009. Sturmian words with balanced construction; 12 pp. In Proceedings of Words 2009, the 7th International Conference on Words (Salerno, Italy), 14–18 September 2009. http://words2009.dia.unisa.it/accepted.html. Reprints were made with permission from the publishers. Contents 1 Introduction ....................................... 9 1.1 Ournumber-theoreticaltools........................ 11 1.1.1 A brief introduction to continued fractions (CF) . 11 1.1.2 TheGaussmap.............................. 14 1.1.3 TheStern–Brocottree......................... 15 1.2 Digitalgeometry................................. 18 1.2.1 Digital lines . 19 1.2.2 CF-based descriptions of digital lines . 24 1.3 Combinatorics on words . 25 1.3.1 Sturmianwords.............................. 27 1.3.2 CF-based descriptions of Sturmian words . 33 1.3.3 Fixed-pointtheoremsforwords.................. 35 2 Summary of papers . 39 2.1 PaperI ........................................ 39 2.2 PaperII........................................ 40 2.3 PaperIII....................................... 41 2.4 PapersIVandV................................. 42 2.5 PaperVI....................................... 45 3 Sammanfattning på svenska . 47 4 Acknowledgments ................................... 51 Bibliography . 53 1. Introduction This thesis is based on a number of different domains of mathematics. The two main domains are digital geometry and combinatorics on words. The problems we treat in this thesis appear also in symbolic dynamics, crystallography, and astronomy. This interdisciplinary character can be both very frustrating and very rewarding. The usual reason for frustration is that you can be surprised that the problem you are working on has already been solved in another domain than your own. You must check a large number of disciplines of natural science to ascertain that your work is original. The rewarding aspect is the fact that once you have managed to formulate and solve a really interesting problem, it can be applied in many different ways, which gives an enormous sense of satisfaction. In this introduction we would like to sketch the problem, present the different terms associated with it, and describe the circumstances in which it appears. Before we present the tools we have used in this thesis and the domains we were working in, we will introduce some terms related to the problem of interest, i.e., to descriptions of the sequence (na)n∈N for a positive irrational a less than 1. These are: 1. the β-sequence defined by a (Bernoulli, Markov, Venkov) β(n)=(n +1)a−na; see Nillsen et al. (1999) [66], 2. the Beatty sequence associated with a: Ba =(an)n∈N+ ;seeBeatty (1926) [6], de Bruijn (1989) [25], Komatsu (1995) [57], 3. the characteristic word,theupper (lower) mechanical word with slope a; see Definition 6 and formula (1.8) in this thesis, 4. rotation on a circle (Sturmian trajectory defined by a); see Arnoux et al. (1999) [3] and Definition 8 in this thesis, 5. Freeman chain code of the line y = ax; see Freeman (1970) [38], and Figure 1.5 and formula (1.6) in this thesis, 6. the cutting sequence of the line y = ax; see Figure 1.5 in this thesis, 7. the billiard word with slope a; see Borel and Reutenauer (2005) [15] and the brief description in Section 1.3.1 of this thesis. Other terms connected with our problem are: Rauzy rules, standard se- quences, balanced words,wordswithminimal complexity,andChristoffel words. 9 Geometry Combinatorics chain codes digital lines on words mechanical words self-similarity minimal complexity billiard words tessellations balanced aperiodic words cutting sequences Crystallography characteristic words Penrose tiling quasicrystals symbolic trajectories STURMIAN toral rotations ( na )n N Symbolic dynamics Beatty sequences zeros of solution Number of a Sturm-Liouville equation theory b - sequences Differential equations Astronomy Figure 1.1: The main domains and terms related to our problem. In Section 1.1 we will give a brief introduction to the number- theoretical tools which we will use in this thesis. In Figure 1.1 we present the main domains and terms related to our problem of describing the sequence (na)n∈N for a ∈ ]0, 1[ \ Q. In Sections 1.2 and 1.3 we will try to link the above mentioned terms to each other. Examples of other places in the literature where the interdisciplinary character of our problem is illuminated are Stolarsky (1976) [80], Bruck- stein (1991) [24], Lothaire (2002) [60, pp. 45–60], Pytheas Fogg (2002) [69, pp. 143–198], Berthé et al. (2005) [12], Berthé (2009) [11], and Harris and Reingold (2004) [44]. 10 1.1 Our number-theoretical tools 1.1.1 A brief introduction to continued fractions (CF) The history of the use of continued fractions (CF) is as long as the his- tory of the use of Euclid’s algorithm (ca 300 BC), because the process of finding the greatest common divisor for two natural positive numbers n n and k is the same process as calculating the CF-expansion of k .A list of important dates and names in the history of CFs can be found in Wikipedia (http://en.wikipedia.org/wiki/Continued_fraction). Here we will only mention Aryabhatta (499), Rafael Bombelli (1579), Pietro Cataldi (1613)—first notation for CFs, John Wallis (1695)—introduction of the term continued fraction, Leonard Euler, Johann Lambert, Joseph Louis Lagrange, Karl Friedrich Gauss, and Bill Gosper (1972) [42]—first exact algorithms for CF-arithmetic. For more historical information see Brezinski (1991) [18], Flajolet et al. (2000) [36], and Vardi (1998) [86]. To illustrate the first statement of this section with an example, we will run Euclid’s algorithm for the numbers 17 and 31: 31 = 1 · 17 + 14 17 = 1 · 14 + 3 14 = 4 · 3+2 3=1 · 2+1 2=2 · 1 and compare it with the following: 17 1 1 1 1 1 = = = = = 31 31 1+ 14 1 1 1 17 17 1+ 1+ 1+ 17 3 1 14 1+ 14 1+ 14 3 1 1 1 = = = . 1 1 1 1+ 1+ 1 + 1 1 1 1+ 1+ 1 + 4+ 2 1 1 3 4+ 4 + 3 1 2 1 + 2 We notice the numbers 1, 1, 4, 1, 2 (in boldface in both calculations), which are the integer parts of the quotients in the performed divisions, appear in both operations. After this first example we will formally define a CF-expansion of a number. We will do this for irrational numbers, because they are at the absolute center of our attention in this thesis. The algorithm for rational numbers is the same as for irrational numbers, but with the difference that it ends after a certain number of steps. 11 Let a be an irrational number. The following algorithm gives the regular (or simple) continued fraction (CF) for a: 1 a + =[a ; a ,a ,a ,...]. 0 1 0 1 2 3 a + 1 1 a + 2 1 a + 3 ··· We define a sequence of integers (an) and a sequence of real numbers (αn) by 1 α0 = a; an = αn and αn+1 = for n ≥ 0. αn − an Then an ≥ 1 and αn > 1 for n ≥ 1. The natural numbers a0,a1,a2,a3,..
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