On the subword complexity of Thue–Morse polynomial extractions

Let the (subword) complexity of a sequence over a finite set Σ be the function , where is the number of distinct blocks of length m in . Let denote the Thue–Morse sequence. In this paper we study the complexity of the sequences , when H(n)∈Q[n] is a polynomial with H(N)⊆N. In particular, we solve an open problem of Allouche and Shallit regarding . We also study the vector space over Z/2Z, spanned by the sequences .