Addition Automata and Attractors of Digit Systems Corresponding to Expanding Rational Matrices
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| Format: | Preprint |
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2025
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| _version_ | 1866913932438929408 |
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| author | Cruz, Anjelo Gabriel R. Loquias, Manuel Joseph C. Thuswaldner, Jörg M. |
| author_facet | Cruz, Anjelo Gabriel R. Loquias, Manuel Joseph C. Thuswaldner, Jörg M. |
| contents | Let $A$ be an expanding $2 \times 2$ matrix with rational entries and $\mathbb{Z}^2[A]$ be the smallest $A$-invariant $\mathbb{Z}$-module containing $\mathbb{Z}^2$. Let $\mathcal{D}$ be a finite subset of $\mathbb{Z}^2[A]$ which is a complete residue system of $\mathbb{Z}^2[A]/A\mathbb{Z}^2[A]$. The pair $(A,\mathcal{D})$ is called a {\em digit system} with {\em base} $A$ and {\em digit set} $\mathcal{D}$. It is well known that every vector $x \in \mathbb{Z}^2[A]$ can be written uniquely in the form \[ x = d_0 + Ad_1 + \cdots + A^kd_k + A^{k+1}p, \] with $k\in \mathbb{N}$ minimal, $d_0,\dots,d_k \in \mathcal{D}$, and $p$ taken from a finite set of {\em periodic elements}, the so-called {\em attractor} of $(A,\mathcal{D})$. If $p$ can always be chosen to be $0$ we say that $(A,\mathcal{D})$ has the {\em finiteness property}.
In the present paper we introduce finite-state transducer automata which realize the addition of the vectors $\pm(1,0)^\top$ and $\pm(0,1)^\top$ to a given vector $x\in \mathbb{Z}^2[A]$ in a number system $(A,\mathcal{D})$ with collinear digit set. These automata are applied to characterize all pairs $(A,\mathcal{D})$ that have the finiteness property and, more generally, to characterize the attractors of these digit systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_06158 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Addition Automata and Attractors of Digit Systems Corresponding to Expanding Rational Matrices Cruz, Anjelo Gabriel R. Loquias, Manuel Joseph C. Thuswaldner, Jörg M. Number Theory Formal Languages and Automata Theory 11A63, 68Q45 Let $A$ be an expanding $2 \times 2$ matrix with rational entries and $\mathbb{Z}^2[A]$ be the smallest $A$-invariant $\mathbb{Z}$-module containing $\mathbb{Z}^2$. Let $\mathcal{D}$ be a finite subset of $\mathbb{Z}^2[A]$ which is a complete residue system of $\mathbb{Z}^2[A]/A\mathbb{Z}^2[A]$. The pair $(A,\mathcal{D})$ is called a {\em digit system} with {\em base} $A$ and {\em digit set} $\mathcal{D}$. It is well known that every vector $x \in \mathbb{Z}^2[A]$ can be written uniquely in the form \[ x = d_0 + Ad_1 + \cdots + A^kd_k + A^{k+1}p, \] with $k\in \mathbb{N}$ minimal, $d_0,\dots,d_k \in \mathcal{D}$, and $p$ taken from a finite set of {\em periodic elements}, the so-called {\em attractor} of $(A,\mathcal{D})$. If $p$ can always be chosen to be $0$ we say that $(A,\mathcal{D})$ has the {\em finiteness property}. In the present paper we introduce finite-state transducer automata which realize the addition of the vectors $\pm(1,0)^\top$ and $\pm(0,1)^\top$ to a given vector $x\in \mathbb{Z}^2[A]$ in a number system $(A,\mathcal{D})$ with collinear digit set. These automata are applied to characterize all pairs $(A,\mathcal{D})$ that have the finiteness property and, more generally, to characterize the attractors of these digit systems. |
| title | Addition Automata and Attractors of Digit Systems Corresponding to Expanding Rational Matrices |
| topic | Number Theory Formal Languages and Automata Theory 11A63, 68Q45 |
| url | https://arxiv.org/abs/2507.06158 |