### A new class of continued fraction expansions

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Let $x\in ]0,1]$ 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...

We give the relationship between regular continued fractions and Lehner fractions, using a procedure known as insertion}. Starting from the regular continued fraction expansion of any real irrational x, when the maximal number of insertions is applied one obtains the Lehner fraction of x. Insertions (and singularizations) show how these (and other) continued fraction expansions are related. We also investigate the relation between Lehner fractions and the Farey expansion (also known as the full...

In this paper the Engel continued fraction (ECF) expansion of any $x\in (0,1)$ is introduced. Basic and ergodic properties of this expansion are studied. Also the relation between the ECF and F. Ryde’s (MNK) is studied.

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