Partial quotients and representation of rational numbers

Abstract

It is shown that there is an absolute constant $C$ such that any rational $\frac{b}{q}\in ]0,1[,(b,q)=1$, admits a representation as a finite sum $\frac{b}{q}={\sum }_{\alpha }\frac{b_{\alpha }}{q_{\alpha }}$ where ${\sum }_{\alpha }{\sum }_i{a}_i(\frac{b_{\alpha }}{q_{\alpha }})<C\log q$ and $\{a_i(x)\}$ denotes the sequence of partial quotients of $x$.