It is well known that every $x\in (0,1]$ can be expanded to an infinite Lüroth series in the form of $$x=\frac{1}{d_1\left(x\right)}+\cdots +\frac{1}{d_1\left(x\right)(d_1\left(x\right)-1)\cdots d_{n-1}\left(x\right)(d_{n-1}\left(x\right)-1)d_n\left(x\right)}+\cdots ,$$ where $d_n\left(x\right)\ge 2$ for all $n\ge 1$. In this paper, sets of points with some restrictions on the digits in Lüroth series expansions are considered. Mainly, the Hausdorff dimensions of the Cantor sets $$F_{\phi }=\{x\in (0,1]:d_n\left(x\right)\ge \phi \left(n\right),\forall n\ge 1\}$$ are completely determined, where $\phi$ is an integer-valued function defined on $\mathbb{N}$, and $\phi \left(n\right)\to \infty$ as $n\to \infty$.

Let ${fₙ}_{n\ge 1}$ be an infinite iterated function system on [0,1] satisfying the open set condition with the open set (0,1) and let Λ be its attractor. Then to any x ∈ Λ (except at most countably many points) corresponds a unique sequence ${aₙ\left(x\right)}_{n\ge 1}$ of integers, called the digit sequence of x, such that $x=li{m}_{n\to \infty }{f}_{a₁\left(x\right)}\circ \cdots \circ f_{aₙ\left(x\right)}\left(1\right)$. We investigate the growth speed of the digits in a general infinite iterated function system. More precisely, we determine the dimension of the set $x\in \Lambda :aₙ\left(x\right)\in B\left(\forall n\ge 1\right),li{m}_{n\to \infty }aₙ\left(x\right)=\infty$ for any infinite subset B ⊂ ℕ, a question posed by Hirst for continued...

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 the two dimensional real vector space ${ℝ}^2$ one can define analogs of the well-known $q$-adic number systems. In these number systems a matrix $M$ plays the role of the base number $q$. In the present paper we study the so-called fundamental domain $ℱ$ of such number systems. This is the set of all elements of ${ℝ}^2$ having zero integer part in their “$M$-adic” representation. It was proved by Kátai and Környei, that $ℱ$ is a compact set and certain translates of it form a tiling of the ${ℝ}^2$. We construct points, where...

Continuous transformations preserving the Hausdorff-Besicovitch dimension (“DP-transformations”) of every subset of R 1 resp. [0, 1] are studied. A class of distribution functions of random variables with independent s-adic digits is analyzed. Necessary and sufficient conditions for dimension preservation under functions which are distribution functions of random variables with independent s-adic digits are found. In particular, it is proven that any strictly increasing absolutely continuous distribution...