### A note on the approximation by continued fractions under an extra condition.

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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...

Let K ⊆ ℝ be the unique attractor of an iterated function system. We consider the case where K is an interval and study those elements of K with a unique coding. We prove under mild conditions that the set of points with a unique coding can be identified with a subshift of finite type. As a consequence, we can show that the set of points with a unique coding is a graph-directed self-similar set in the sense of Mauldin and Williams (1988). The theory of Mauldin and Williams then provides a method...

Let $\beta >1$ be a non-integer. We consider expansions of the form ${\sum}_{i=1}^{\infty}{d}_{i}/{\beta}^{i}$, where the digits ${\left({d}_{i}\right)}_{i\ge 1}$ are generated by means of a Borel map ${K}_{\beta}$ defined on ${\{0,1\}}^{\mathbb{N}}\times [0,\lfloor \beta \rfloor \left(\beta -1\right)]$. We show existence and uniqueness of a ${K}_{\beta}$-invariant probability measure, absolutely continuous with respect to ${m}_{p}\otimes \lambda $, where ${m}_{p}$ is the Bernoulli measure on ${\{0,1\}}^{\mathbb{N}}$ with parameter $p$ ($0<p<1$) and $\lambda $ is the normalized Lebesgue measure on $[0,\lfloor \beta \rfloor (\beta -1\left)\right]$. Furthermore, this measure is of the form ${m}_{p}\otimes {\mu}_{\beta ,p}$, where ${\mu}_{\beta ,p}$ is equivalent to $\lambda $. We prove that the measure of maximal entropy and ${m}_{p}\otimes \lambda $ are mutually singular. In...

Let $\beta >1$ be a non-integer. We consider $\beta $-expansions of the form ${\sum}_{i=1}^{\infty}{d}_{i}/{\beta}^{i}$, where the digits ${\left({d}_{i}\right)}_{i\ge 1}$ are generated by means of a Borel map ${K}_{\beta}$ defined on ${\{0,1\}}^{\mathbb{N}}\times [0,\lfloor \beta \rfloor /\left(\beta -1\right)]$. We show that ${K}_{\beta}$ has a unique mixing measure ${\nu}_{\beta}$ of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure ${\nu}_{\beta}$ the digits ${\left({d}_{i}\right)}_{i\ge 1}$ form a uniform Bernoulli process. In case 1 has a finite greedy expansion with positive coefficients, the measure of maximal entropy is Markov. We also discuss the uniqueness of $\beta $-expansions....

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