On the arithmetical complexity of Sturmian words

  • Authors:

  • J. Cassaigne;A. E. Frid

  • Affiliations:

  • Institut de mathématiques de Luminy Case 907, 163 av. de Luminy, F-13288 Marseille Cedex 9, France;Sobolev Institute of Mathematics SB RAS, Koptyug av., 4, 630090 Novosibirsk, Russia

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2007

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Abstract

Using the geometric dual technique by Berstel and Pocchiola, we give a uniform O(n^3) upper bound for the arithmetical complexity of a Sturmian word. We also give explicit expressions for the arithmetical complexity of Sturmian words of slope between 1/3 and 2/3 (in particular, of the Fibonacci word). In this case, the difference between the genuine arithmetical complexity function and our upper bound is bounded, and ultimately 2-periodic. In fact, our formula is valid not only for Sturmian words but for rotation words from a wider class.