On the characterization of Peth\H{o}'s Loudspeaker

TL;DR

This paper investigates the structure of Peth ext{o}'s Loudspeaker, a complex set related to Gaussian shift radix systems, providing a conjecture, partial proofs, and geometric measurements under certain assumptions.

Contribution

It formulates and proves parts of a conjecture characterizing Peth ext{o}'s Loudspeaker, including algorithmic bounds and geometric properties.

Findings

Containment of the Loudspeaker in a conjectured set

Algorithmic approximation of the set's boundary

Computed circumference and area assuming the conjecture

Abstract

For $d \in \mathbb{N}$ and $\mathbf{r} \in \mathbb{C}^d$ let $\gamma_\mathbf{r}: \mathbb{Z}[\mathrm{i}]^d \to \mathbb{Z}[\mathrm{i}]^d$, where $\gamma_\mathbf{r}(\mathbf{a})=(a_2,...,a_d,$ $-\lfloor\mathbf{r}\mathbf{a}\rfloor)$ for $\mathbf{a}=(a_1,...,a_d)$, denote the (d-dimensional) Gaussian shift radix system associated with $\mathbf{r}$. $\gamma_\mathbf{r}$ is said to have the finiteness property iff all orbits of $\gamma_\mathbf{r}$ end up in $(0,...,0)$; the set of all corresponding $\mathbf{r} \in \mathbb{C}^d$ is denoted by $\mathcal{G}_{d}^{(0)}$. It has a very complicated structure even for $d=1$. In the present paper a conjecture on the full characterization of $\mathcal{G}_{1}^{(0)}$ - which is known as Peth\H{o}'s Loudspeaker - is formulated and proven in substantial parts. It is shown that $\mathcal{G}_{1}^{(0)}$ is contained in a conjectured characterizing set $\mathcal{G}_C$. The other inclusion is settled algorithmically for large regions leaving only small areas of uncertainty. Furthermore the circumference and area of the Loudspeaker are computed under the assumption that the conjecture holds. The proven parts of the conjecture also allow to fully identify all so-called critical and weakly critical points of $\mathcal{G}_{1}^{(0)}$.