Geodesic
On the sequence of the first binary digits of the fractional parts of the values of a polynomial
Čebyševskij sbornik, Tome 22 (2021) no. 1, pp. 482-487

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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 $k$.
Keywords: Combinatorics on words, symbolical dynamics, unipotent torus transformation, Weiyl lemma. 
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     author = {A. Ya. Belov and G. V. Kondakov and I. V. Mitrofanov and M. M. Golafshan},
     title = {On the sequence of the first binary digits of the fractional parts of the values of a polynomial},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {482--487},
     publisher = {mathdoc},
     volume = {22},
     number = {1},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2021_22_1_a32/}
}
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A. Ya. Belov; G. V. Kondakov; I. V. Mitrofanov; M. M. Golafshan. On the sequence of the first binary digits of the fractional parts of the values of a polynomial. Čebyševskij sbornik, Tome 22 (2021) no. 1, pp. 482-487. http://geodesic.mathdoc.fr/item/CHEB_2021_22_1_a32/