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This question is (remotely) related to How to find a "simple" fraction between two other fractions?, but is not answered in that older post.

Let $f_1=\frac{a}{b}$ and $f_2=\frac{c}{d}$ be two reduced fractions with $bc-ad > 1$ (and hence $\frac{a}{b} \lt \frac{c}{d}$) and $a,b,c,d$ positive. Then the theory of Farey sequences tells us that in-between $f_1$ and $f_2$ we may find a reduced fraction $\frac{x}{y}$ with low denominator, i.e. $y \leq {\rm max}(b,d)$. We know also that if we take mediants iteratively between $f_1$ and $f_2$, we eventually reconstruct the whole part of the Stern-Brocot tree between $f_1$ and $f_2$, so that we will eventually encounter fractions with low denominator.

Formally, let $T$ be the following transform on finite increasing sequences of reduced fractions :

$$ T\bigg(\frac{a_1}{b_1}\lt\frac{a_2}{b_2}\lt\frac{a_3}{b_3} \lt \ldots\lt\frac{a_n}{b_n}\bigg)= \bigg(\frac{a_1}{b_1}\lt\frac{a_1+a_2}{b_1+b_2}\lt\frac{a_2}{b_2}\lt\frac{a_2+a_3}{b_2+b_3}\lt \frac{a_3}{b_3} \lt \ldots\lt\frac{a_n}{b_n}\bigg) $$

so that $T$ transforms a sequence of length $n$ into a larger sequence, of length $2n+1$.

Then some iterate $T^N\bigg({\frac{a}{b}<\frac{c}{d}}\bigg)$ contains a fraction with low denominator. How fast is that fraction reached ? I am expecting two different sorts of answers :

  • Find a bound $N(m)$, such that one always finds a fraction with low denominator in $N(m)$ steps in the worst case (in terms of $m={\sf max}(b,d)$).

  • Find a bound $B(m)$ on the size of the denominators encountered before finding a low denominator.

    For example, I have computed that $N(10)=8$ and $B(10)=327$ (corresponding to the two worst cases $\frac{1}{10} < \frac{8}{9}$ and $\frac{1}{9} < \frac{9}{10}$ ).

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Ewan Delanoy
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(Je préfèrerais donner une réponse en français, mais ça ne serait pas aimable vis à vis des éventuels lecteurs anglophones.)

I founded a bound but it does not depend only on $m$ but on $bc-ad$. It is quite simple (and I won't be able to give a complex solution because of my young and limited knowledge) :

I expressed $\left(\begin{array}{c} y\\ x\end{array}\right)$ in the basis $\left(\left(\begin{array}{c} b\\ a\end{array}\right),\left(\begin{array}{c} d\\ c\end{array}\right)\right)$.

I obtained $\dfrac{\alpha}{bc-ad}\left(\begin{array}{c} b\\ a\end{array}\right)+\dfrac{\beta}{bc-ad}\left(\begin{array}{c} d\\ c\end{array}\right)=\left(\begin{array}{c} y\\ x\end{array}\right)$ so $\alpha\left(\begin{array}{c} b\\ a\end{array}\right)+\beta\left(\begin{array}{c} d\\ c\end{array}\right)=(bc-ad)\left(\begin{array}{c} y\\ x\end{array}\right)$

and $y\leq m$ so $x\leq \dfrac{mc}{d}$ and $x\geq \dfrac{ma}{b}$

if we note $n=min(b,d)$.

$\alpha\leq \dfrac{m}{n}(bc-ad)$ and $\beta\leq \dfrac{m}{n}(bc-ad)$.

Finally $N\leq max(\alpha,\beta)\leq \dfrac{m}{n} (bc-ad)$.

Assuming that the fractions are lower than $1$, $bc-ad\leq mn$.

$N(m)\leq m^2$

$B(m)\leq\mathcal{F}_{m^{2}}m$

where $\mathcal{F}$ is the fibonacci sequence. This is not wonderful but this is a start.

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