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Let $x \in [0, 1]$ be irrational and let $$x = \cfrac1{a_1 + \cfrac1{a_2 + \cfrac1{a_3 + \frac1\ddots}}}$$ be the regular continued fraction expansion of $x$. If you truncate the continued fraction expansion at $n$ you get the $n$th convergent $A_n/B_n$ to $x$.

Let $R_x(n) = x - A_n/B_n$. I know that $$R_x(n) \leq \frac1{B_n B_{n+1}}$$ For $\phi = \frac{\sqrt{5} - 1}{2}$ we have that $R_\phi(n) = O(\phi^{2n})$ and this holds for any $x$ so it is the worst-case approximation to the $R_x$.

Are there any such approximation for the "average-case"?

EDIT: I don't have a rigorous definition of "average-case". But I would be interested in approximations that work for almost all $x$, for example.

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Lochs' Theorem implies, that

$$R(n) = \Theta \left(\exp\left(- \frac{\pi^2 n}{6 \ln 3}\right)\right)$$

for almost all $x \in \mathbb{R}$.

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  • $\begingroup$ Should there be an $n$ in there somewhere? $\endgroup$ – Antonio Vargas Nov 2 '16 at 11:32
  • $\begingroup$ @AntonioVargas yup you're quite right $\endgroup$ – 0x539 Nov 2 '16 at 11:34

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