Lengths of continued fractions for the numbers with fixed ratio - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T14:04:15Z http://mathoverflow.net/feeds/question/41557 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41557/lengths-of-continued-fractions-for-the-numbers-with-fixed-ratio Lengths of continued fractions for the numbers with fixed ratio Alexey Ustinov 2010-10-09T02:15:03Z 2010-10-09T13:06:10Z <p>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. </p> <p>Is this result new?</p> http://mathoverflow.net/questions/41557/lengths-of-continued-fractions-for-the-numbers-with-fixed-ratio/41589#41589 Answer by Fedor Petrov for Lengths of continued fractions for the numbers with fixed ratio Fedor Petrov 2010-10-09T13:06:10Z 2010-10-09T13:06:10Z <p>It definitely is not new for the length, and I am nearly sure that is not for height either.</p> <p>See, for example, </p> <p>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.</p>