Non-Expansive Matrix Based number Systems
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913094889897984 |
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| author | Blažek, Adam Hare, Kevin G. Pelantová, Edita |
| author_facet | Blažek, Adam Hare, Kevin G. Pelantová, Edita |
| contents | Let $M = \left(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right)$ be a $2 \times 2$ Jordan block with eigenvalue $1$, and let $\mathcal{D} = \{\left(\begin{smallmatrix}0 \\ 1 \end{smallmatrix}\right), \left(\begin{smallmatrix} 0 \\ -1 \end{smallmatrix} \right)\}$. In this paper, we answer a question of Caldwell, Hare, and Vávra about the minimal length representation of $\left( \begin{smallmatrix} a \\ b \end{smallmatrix} \right) = \sum_{i=0}^{k-1} M^i d_i$ with $d_i \in \mathcal{D}$. Further, we extend the work of Caldwell, Hare, and Vávra to consider the case of $n \times n$ Jordan blocks with eigenvalue $-1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_04990 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-Expansive Matrix Based number Systems Blažek, Adam Hare, Kevin G. Pelantová, Edita Number Theory 11K16, 11C20, 15B36 Let $M = \left(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right)$ be a $2 \times 2$ Jordan block with eigenvalue $1$, and let $\mathcal{D} = \{\left(\begin{smallmatrix}0 \\ 1 \end{smallmatrix}\right), \left(\begin{smallmatrix} 0 \\ -1 \end{smallmatrix} \right)\}$. In this paper, we answer a question of Caldwell, Hare, and Vávra about the minimal length representation of $\left( \begin{smallmatrix} a \\ b \end{smallmatrix} \right) = \sum_{i=0}^{k-1} M^i d_i$ with $d_i \in \mathcal{D}$. Further, we extend the work of Caldwell, Hare, and Vávra to consider the case of $n \times n$ Jordan blocks with eigenvalue $-1$. |
| title | Non-Expansive Matrix Based number Systems |
| topic | Number Theory 11K16, 11C20, 15B36 |
| url | https://arxiv.org/abs/2605.04990 |