Canonical number systems in the ring of gaussian integers $\mathbb{Z}\left[i\right]$ are the natural generalization of ordinary $q$-adic number systems to $\mathbb{Z}\left[i\right]$. It turns out, that each gaussian integer has a unique representation with respect to the powers of a certain base number $b$. In this paper we investigate the sum of digits function ${\nu }_b$ of such number systems. First we prove a theorem on the sum of digits of numbers, that are not divisible by the $f$-th power of a prime. Furthermore, we establish an Erdös-Kac type theorem...

For any $x\in (0,1]$, let $$x=\frac{1}{d_1}+\frac{1}{d_1(d_1-1)d_2}+\cdots +\frac{1}{d_1(d_1-1)\cdots d_{n-1}(d_{n-1}-1)d_n}+\cdots$$ be its Lüroth expansion. Denote by $P_n\left(x\right)/Q_n\left(x\right)$ the partial sum of the first $n$ terms in the above series and call it the $n$th convergent of $x$ in the Lüroth expansion. This paper is concerned with the efficiency of approximating real numbers by their convergents ${\{P_n\left(x\right)/Q_n\left(x\right)\}}_{n\ge 1}$ in the Lüroth expansion. It is shown that almost no points can have convergents as the optimal approximation for infinitely many times in the Lüroth expansion. Consequently, Hausdorff dimension is introduced to quantify the set of...

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...

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...