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...

By using a generating function approach it is shown that the sum-of-digits function (related to specific finite and infinite linear recurrences) satisfies a central limit theorem. Additionally a local limit theorem is derived.

It is known that with a non-unit Pisot substitution $\sigma$ one can associate certain fractal tiles, so-called Rauzy fractals. In our setting, these fractals are subsets of a certain open subring of the adèle ring of the associated Pisot number field. We present several approaches on how to define Rauzy fractals and discuss the relations between them. In particular, we consider Rauzy fractals as the natural geometric objects of certain numeration systems, in terms of the dual of the one-dimensional realization...

The present paper deals with the summatory function of functions acting on the digits of an $q$-ary expansion. In particular let $n$ be a positive integer, then we call$$\begin{array}{c}n=\sum _{r=0}^{\ell }{d}_r\left(n\right)q^r\text{with}{d}_r\left(n\right)\in \{0,...,q-1\}\end{array}$$its $q$-ary expansion. We call a function $f$strictly $q$-additive, if for a given value, it acts only on the digits of its representation, i.e.,$$f\left(n\right)=\sum _{r=0}^{\ell }f\left(d_r\left(n\right)\right).$$Let $p\left(x\right)={\alpha }_0{x}^{{\beta }_0}+\cdots +{\alpha }_d{x}^{{\beta }_d}$ with ${\alpha }_0,{\alpha }_1,...,{\alpha }_d,\in ℝ$, ${\alpha }_0\>0$, ${\beta }_0\>\cdots \>{\beta }_d\ge 1$ and at least one ${\beta }_i\notin \mathbb{Z}$. Then we call $p$ a pseudo-polynomial.The goal is to prove that for a $q$-additive function $f$ there exists an $\epsilon \>0$ such that$$\begin{array}{c}\sum _{n\le N}f\left(⌊p\left(n\right)⌋\right)={\mu }_{f}N{log}_q\left(p\left(N\right)\right)\hfill \\ \hfill +N{F}_{f,{\beta }_0}\left({log}_q\left(p\left(N\right)\right)\right)+𝒪\left(N^{1-\epsilon }\right),\end{array}$$where ${\mu }_f$ is the mean of the values of $f$...

The aim of this work is to estimate exponential sums of the form ${\sum }_{n\le x}\Lambda \left(n\right)exp\left(2i\pi (f\left(n\right)+\beta n)\right)$, where Λ denotes von Mangoldt’s function, f a digital function, and β ∈ ℝ a parameter. This result can be interpreted as a Prime Number Theorem for rotations (i.e. a Vinogradov type theorem) twisted by digital functions.

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...