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A question on continued fraction expansion

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A question on continued fraction expansion

Let $\alpha_j=G^j(\alpha)$ is the j^th fractional part in the continued fraction expansion , where $G$ is the Gause map: $x\map {1/x}=1/x-[1/x]$. If there is such fact that the product $\alpha_{n-1}\cdot\alpha_{n-j}$ is no large than $2^{-j/2}$, for any ( or sufficiently large) $n,j$?


Let $\alpha_j=G^j(\alpha)$ is the j^the fractional part in the continued fraction expansion , where $G$ is the Gause map: $x\map {1/x}=1/x-[1/x]$. If there is such fact that the product $\alpha_{n-1}\cdot\alpha_{n-j}$ is no large than $2^{-j/2}$, for any ( or sufficiently large) $n,j$.

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