On Arithmetic Index in the Generalized Thue-Morse Word

  • Olga G. ParshinaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10432)


Let q be a positive integer. Consider an infinite word \(\omega =w_0w_1w_2\cdots \) over an alphabet of cardinality q. A finite word u is called an arithmetic factor of \(\omega \) if \(u=w_cw_{c+d}w_{c+2d}\cdots w_{c+(|u|-1)d}\) for some choice of positive integers c and d. We call c the initial number and d the difference of u. For each such u we define its arithmetic index by \(\lceil \log _q d\rceil \) where d is the least positive integer such that u occurs in \(\omega \) as an arithmetic factor with difference d. In this paper we study the rate of growth of the arithmetic index of arithmetic factors of a generalization of the Thue-Morse word defined over an alphabet of prime cardinality. More precisely, we obtain upper and lower bounds for the maximum value of the arithmetic index in \(\omega \) among all its arithmetic factors of length n.


Arithmetic index Arithmetic progression Thue-Morse word 


  1. 1.
    Allouche, J.P., Shallit, J.: The ubiquitous Prouhet-Thue-Morse sequence. In: Ding, C., Helleseth, T., Niederreiter, H. (eds.) Sequences and their Applications. Discrete Mathematics and Theoretical Computer Science, pp. 1–16. Springer, London (1999)Google Scholar
  2. 2.
    Avgustinovich, S.V., Fon-Der-Flaass, D.G., Frid, A.E.: Arithmetical complexity of infinite words. In: Words, Languages and Combinatorics III, Kyoto 2000, pp. 51–62. World Science Publisher, River Edge (2003). doi: 10.1142/9789812704979_0004
  3. 3.
    Avgustinovich, S.V., Cassaigne, J., Frid, A.E.: Sequences of low arithmetical complexity. Theor. Inform. Appl. 40(4), 569–582 (2006). doi: 10.1051/ita:2006041 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brlek, S.: Enumeration of factors in the Thue-Morse word. Discrete Appl. Math. 24(1–3), 83–96 (1989). doi: 10.1016/0166-218X(92)90274-E. First Montreal Conference on Combinatorics and Computer Science (1987)
  5. 5.
    Cassaigne, J., Frid, A.E.: On the arithmetical complexity of Sturmian words. Theoret. Comput. Sci. 380(3), 304–316 (2007). doi: 10.1016/j.tcs.2007.03.022 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ehrenfeucht, A., Lee, K.P., Rozenberg, G.: Subword complexities of various classes of deterministic developmental languages without interactions. Theor. Comput. Sci. 1(1), 59–75 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Frid, A.E.: Sequences of linear arithmetical complexity. Theor. Comput. Sci. 339(1), 68–87 (2005). doi: 10.1016/j.tcs.2005.01.009 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Frid, A.E.: On possible growths of arithmetical complexity. Theor. Inform. Appl. 40(3), 443–458 (2006). doi: 10.1051/ita:2006021 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    de Luca, A., Varricchio, S.: Some combinatorial properties of the Thue-Morse sequence and a problem in semigroups. Theor. Comput. Sci. 63(3), 333–348 (1989). doi: 10.1016/0304-3975(89)90013-3 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Parshina, O.G.: On arithmetic progressions in the generalized Thue-Morse word. In: Manea, F., Nowotka, D. (eds.) WORDS 2015. LNCS, vol. 9304, pp. 191–196. Springer, Cham (2015). doi: 10.1007/978-3-319-23660-5_16 CrossRefGoogle Scholar
  11. 11.
    Thue, A.: Über die gegenseitige lage gleicher teile gewisser zeichenreichen. Skr. Vid.-Kristiana I. Mat. Naturv. Klasse 1, 1–67 (1912)Google Scholar
  12. 12.
    Tromp, J., Shallit, J.: Subword complexity of a generalized Thue-Morse word. Inform. Process. Lett. 54(6), 313–316 (1995). doi: 10.1016/0020-0190(95)00074-M MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Van der Waerden, B.: Beweis einer baudetschen vermutung. Nieuw Arch. Wisk. 15, 212–216 (1927)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Sobolev Institute of Mathematic SB RASNovosibirskRussia
  2. 2.Institut Camille JordanUniversité de Lyon, Université Claude Bernard Lyon 1Villeurbanne CedexFrance

Personalised recommendations