Exercise 6.2.3 (Non reduced fractions solutions of (6.2))

Question

Suppose $h = a$, $k = b$ is a solution of the inequality (6.2) for some irrational $\xi$. Prove that only a finite number of pairs $h,k$ in the set $(h = ma,\ k=mb;\ m = 1,2,3,\ldots)$ satisfy (6.2).

Richard Ganaye · Accepted Answer

ewcommand{\N}{\mathbb{N}}

\begin{proof} 
By hypothesis
$$\left| \xi - \frac{a}{b} ight| < \frac{1}{\sqrt{5}\,  b^2}.$$
If $h = ma,\ k =mb$ for some $m \in \N^*$, then
\begin{align*}
\left| \xi - \frac{h}{k} ight| < \frac{1}{\sqrt{5}\,  k^2} & \iff \left| \xi - \frac{ma}{mb} ight| < \frac{1}{\sqrt{5}\,  m^2  b^2}\
&\iff \left| \xi - \frac{a}{b} ight| < \frac{1}{\sqrt{5}\,  m^2  b^2}\
&\iff 0 < m < \left[ \sqrt{5} b^2 \left| \xi -a/b ight| ight]^{-1/2}.
\end{align*}
Hence there are only finitely many $m \in \N^*$ such that $ h = ma,\ k = mb$ satisfy
$$\left| \xi - \frac{h}{k} ight| < \frac{1}{\sqrt{5}\,  k^2} .$$
\end{proof}

Exercise 6.2.3 (Non reduced fractions solutions of (6.2))

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Exercise 6.2.3 (Non reduced fractions solutions of (6.2))

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