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

Abstract

Let 𝑃(𝑛) be a polynomial, having an irrational coefficient of the highest degree. A word 𝑤 (𝑤 = (𝑤𝑛), 𝑛 ∈ N) consists of a sequence of first binary numbers of {𝑃(𝑛)} i.e. 𝑤𝑛 = [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.