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arXiv · 1208.3402

Partial quotients and representation of rational numbers

Jean Bourgain
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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 $x$.

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Jean Bourgain. 2012-08-16. Partial quotients and representation of rational numbers. https://arxiv.org/abs/1208.3402

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