On the subword complexity of Thue–Morse polynomial extractions

  • Authors:

  • Yossi Moshe

  • Affiliations:

  • CNRS, LRI, Université Paris Sud, Orsay, France and Einstein Institute, The Hebrew University, Jerusalem, Israel

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

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Abstract

Let the (subword) complexity of a sequence u=(u"n)"n"="0^~ over a finite set @S be the function m@?P"u(m), where P"u(m) is the number of distinct blocks of length m in u. Let t=(t"n)"n"="0^~ denote the Thue-Morse sequence. In this paper we study the complexity of the sequences t"H=(t"H"("n"))"n"="0^~, when H(n)@?Q[n] is a polynomial with H(N)@?N. In particular, we solve an open problem of Allouche and Shallit regarding (t"n"^"2)"n"="0^~. We also study the vector space over Z/2Z, spanned by the sequences t"H.