# On approximation by Lüroth series

• Volume: 8, Issue: 2, page 331-346
• ISSN: 1246-7405

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## Abstract

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Let $x\in \right]0,1\right]$ and ${p}_{n}/{q}_{n},n\ge 1$ be its sequence of Lüroth Series convergents. Define the approximation coefficients ${\theta }_{n}={\theta }_{n}\left(x\right)$ by ${q}_{n}x-{p}_{n},n\ge 1$. In [BBDK] the limiting distribution of the sequence ${\left({\theta }_{n}\right)}_{n\ge 1}$ was obtained for a.e. $x$ using the natural extension of the ergodic system underlying the Lüroth Series expansion. Here we show that this can be done without the natural extension. In fact we will prove that for each $n,{\theta }_{n}$ is already distributed according to the limiting distribution. Using the natural extension we will study the distribution for a.e. $x$ of the sequence ${\left({\theta }_{n},{\theta }_{n+1}\right)}_{n\ge 1}$ and related sequences like ${\left({\theta }_{n}+{\theta }_{n+1}\right)}_{n\ge 1}$. It turns out that for a.e. $x$ the sequence ${\left({\theta }_{n},{\theta }_{n+1}\right)}_{n\ge 1}$ is distributed according to a continuous singular distribution function G. Furthermore we will see that two consecutive $\theta$’s are positively correlated.

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