Partial quotients and representation of rational numbers

TL;DR

The paper demonstrates that any rational number between 0 and 1 with coprime numerator and denominator can be expressed as a finite sum of fractions, with the sum of partial quotients controlled logarithmically, revealing a new structural property.

Contribution

It introduces a universal bound on the sum of partial quotients in representations of rationals, linking continued fraction properties to rational decompositions.

Findings

Existence of a universal constant C for the representation

Representation bounds involve logarithmic growth in q

Provides a new perspective on rational number decompositions

Abstract

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