Inverse problems of symbolic dimamics

Let $P(n)$ be a polynomial with irrational greatest coefficient. Let also a superword $W$ $(W=(w_n),n\in\mathbb N)$ be the sequence of first binary digits of $\{P(n)\}$, i.e. $w_n=[2\{P(n)\}]$, and $T(k)$ be the number of different subwords of $W$ whose length is equal to $k$. The main result of the paper is the following:
Theorem 1.1. For any $n$ there exists a polynomial $Q(k)$ such that if $deg(P)=n$, then $T(k)=Q(k)$ for all sufficiently large $k$.