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If $\theta$ is irrational with continued fraction representation $[0;a_1,a_2,\ldots]$, $\lbrace \frac{m_k}{n_k} \rbrace$ is the sequence of principal convergents of $\theta$ and $\lbrace b_k\rbrace$ is a sequence of even integers such that $\vert b_k \vert \le \vert a_{k+1} \vert$, how do we get

$$\vert b_k \vert \Vert n_k \theta \Vert \le\frac{a_{k+1}}{n_{k+1}}?$$

I have the recurrence relation $n_{k+1}=a_{k+1}n_k+n_{k-1}$.

Here's what I have: (1) $\frac{a_{k+1}}{n_{k+1}}=\frac{a_{k+1}}{a_{k+1}n_k+n_{k-1}}=\frac{1}{n_k}-\frac{n_{k-1}}{n_k}\cdot n_{k+1},$ but I'm not sure how to involve $\theta.$

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up vote 2 down vote accepted

I assume that $||x||$ is the distance between $x$ and the nearest integer. $\theta$ is between the two successive convergents $m_k/n_k$ and $m_{k+1}/n_{k+1}$, and $$\frac{m_k}{n_k}-\frac{m_{k+1}}{n_{k+1}}=\pm \frac{1}{n_k n_{k+1}}.\qquad (*)$$ Therefore, $$||n_k \theta||\le |m_k - n_k\theta|\le |m_k-n_k\frac{m_{k+1}}{n_{k+1}}|=\frac{1}{n_{k+1}}. $$ The desired result follows immediately.

(*) is standard and follows by induction from the recurrences $$ n_{k+1}=a_{k+1} n_k + n_{k-1}, \qquad m_{k+1} = a_{k+1} m_k + m_{k-1}. $$

Re the comment, if you write $$ \phi_k(x):=a_0+1/(a_1+1/(a_2+1/(\dots+1/(a_k+x)\dots), $$ then $\phi_k$ is monotonic, $\phi_k(0)=m_k/n_k$, $\phi_k(a_{k+1}^{-1})=m_{k+1}/n_{k+1}$, and $\phi_k(\zeta^{-1})=\theta$, where $$ \zeta = a_{k+1}+1/(a_{k+2}+1/(a_{k+3}+\dots>a_{k+1}. $$ Therefore, $0<\zeta^{-1}<a_{k+1}^{-1}$, so by the monotonicity of $\phi_k$, $\theta$ is between $m_k/n_k$ and $m_{k+1}/n_{k+1}$.

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Since $n_k \theta$ can be interpreted as a point on the unit circle, I meant for the $\Vert$ to stand for the norm. Is that consistent with your interpretation? Why is $\theta$ in between those successive convergents? Is that a number theory property? Thanks for your answer. – The Substitute Jan 18 '13 at 2:47
On the unit circle, the usual norm is constant and equal to 1, but I doubt that this is your definition of $||$. So, I need more explanation to understand what you mean by $||$. – David Moews Jan 18 '13 at 3:17
why is $\Vert n_k \theta\Vert \le \vert m_k - n_k \theta\vert $? – The Substitute Apr 15 '13 at 6:35
Assuming that $||x||$ is the distance between $x$ and the nearest integer, this is true because $m_k$ is an integer, and so the distance between $n_k\theta$ and the nearest integer is no bigger than the distance between $m_k$ and $n_k\theta$. – David Moews Apr 15 '13 at 9:47
Since $\theta$ is between $m_k/n_k$ and $m_{k+1}/n_{k+1}$, $|m_k/n_k-\theta|\le |m_k/n_k-m_{k+1}/n_{k+1}|$. Multiplying by $n_k$ then gives $|m_k-n_k \theta|\le |m_k-n_k(m_{k+1}/n_{k+1})|$. – David Moews Apr 15 '13 at 20:04

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