arXiv subjects

Jörg M. Thuswaldner

Publications and source records attributed to Jörg M. Thuswaldner.

21 records · Page 2Linked to original sources

arXiv2014

Characterization of the numbers which satisfy the height reducing property

Let $α$ be a complex number. We show that there is a finite subset $F$ of the ring of the rational integers $\mathbb{Z}$, such that $F\left[ α\right] =\mathbb{Z}\left[ α\right]$, if and only if $α$ is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one, or all of modulus greater than one. This completes the answer to a question, on the numbers satisfying the height reducing property, posed in [3].

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arXiv2013

Shift Radix Systems - A Survey

Let $d\ge 1$ be an integer and ${\bf r}=(r_0,\dots,r_{d-1}) \in \mathbf{R}^d$. The {\em shift radix system} $τ_\mathbf{r}: \mathbb{Z}^d \to \mathbb{Z}^d$ is defined by $$ τ_{\bf r}({\bf z})=(z_1,\dots,z_{d-1},-\lfloor {\bf r} {\bf z}\rfloor)^t \qquad ({\bf z}=(z_0,\dots,z_{d-1})^t). $$ $τ_\mathbf{r}$ has the {\em finiteness property} if each ${\bf z} \in \mathbb{Z}^d$ is eventually mapped to ${\bf 0}$ under iterations of $τ_\mathbf{r}$. In the present survey we summarize results on these nearly linear mappings. We discuss how these mappings are related to well-known numeration systems, to rotations with round-offs, and to a conjecture on periodic expansions w.r.t.\ Salem numbers. Moreover, we review the behavior of the orbits of points under iterations of $τ_\mathbf{r}$ with special emphasis on ultimately periodic orbits and on the finiteness property. We also describe a geometric theory related to shift radix systems.

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arXiv2010

Digit systems over commutative rings

Let $\E$ be a commutative ring with identity and $P\in\E[x]$ be a polynomial. In the present paper we consider digit representations in the residue class ring $\E[x]/(P)$. In particular, we are interested in the question whether each $A\in\E[x]/(P)$ can be represented modulo $P$ in the form $e_0+e_1 X + \cdots + e_h X^h$, where the $e_i\in\E[x]/(P)$ are taken from a fixed finite set of digits. This general concept generalises both canonical number systems and digit systems over finite fields. Due to the fact that we do not assume that $0$ is an element of the digit set and that $P$ need not be monic, several new phenomena occur in this context.

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