On the sequence of the first binary digits of the fractional parts of the values of a polynomial

Let $P(n)$ be a polynomial, having an irrational coefficient of the highest degree. A word $w$ $(w=(w_n), n\in \mathbb{N})$ consists of a sequence of first binary numbers of $\{P(n)\}$ i.e. $w_n=[2\{P(n)\}]$. Denote by $T(k)$ the number of different subwords of $w$ of length $k$ . We'll formulate the main result of this paper.
Theorem. There exists a polynomial $Q(k)$, depending only on the power of the polynomial $P$, such that $T(k)=Q(k)$ for sufficiently great $k$.