12
$\begingroup$

Let $s(x)$ is the length of continued fraction expansion of $x$, and let $l(x)$ be the sum of partial quotients. I can prove that for any rational $\alpha$ ratios $\frac{s(\alpha x)}{s(x)}$ and $\frac{l(\alpha x)}{l(x)}$ (for all rational $x$) are bounded with some constants depending on $\alpha$ only.

Is this result new?

$\endgroup$
3
  • $\begingroup$ I would be really surprised if it wasn't. After all, for l(x) this is just the length of the Euclid algorithm for x=p/q, so if we, say, multiply it by 2/3, then it is just the Euclid algorithm for 2p/3q... must be well known. Perhaps, even Euclid himself knew it. ;) $\endgroup$ Oct 9, 2010 at 10:49
  • $\begingroup$ Yes, it is more or less clear. It is not surprising. It is not hard. But was it already proved or not? $\endgroup$ Oct 9, 2010 at 12:07
  • 1
    $\begingroup$ @Nikita It is not Euclid algorithm for 2p/3q at all :), and all the proofs I know are quite non-trivial, though some of them are respectively short. $\endgroup$ Oct 9, 2010 at 13:10

1 Answer 1

8
$\begingroup$

It definitely is not new for the length, and I am nearly sure that is not for height either.

See, for example,

Labhalla, Salah; Lombardi, Henri Transformation homographique appliqu´ee `a un d´eveloppement en fraction continue fini ou infini. (French) [Fractional linear transformations applied to finite and infinite continued fractions] Acta Arith. 73 (1995), no. 1, 29–41.

$\endgroup$
4
  • $\begingroup$ Thank you Fedor. I'll be greatful for any other refferences concerning this question. $\endgroup$ Oct 9, 2010 at 22:29
  • 1
    $\begingroup$ see also Knuth (Art of Computer Programming), solution of the ex. 4.5.3.15 By the way, I know all these references just because this was a problem proposed there (mathsoc.spb.ru/konkurs/contest10.pdf, how to make normal links here?) (by Elena Golubeva), and students not only proposed their original solutions, but one of them found a lot of references. $\endgroup$ Oct 10, 2010 at 15:40
  • $\begingroup$ Do you have these solutions and references in electronic form? $\endgroup$ Oct 10, 2010 at 22:36
  • $\begingroup$ @Alexey: I sent you smth by e-mail $\endgroup$ Oct 11, 2010 at 7:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.