Let A be a $d\times d$ matrix with rational entries which has no eigenvalue $\lambda \in \mathbb{C}$ of absolute value $\left|\lambda \right|<1$ and let ${\mathbb{Z}}^d[A]$ be the smallest nontrivial A-invariant $\mathbb{Z}$-module. We lay down a theoretical framework for the construction of digit systems $(A,\mathcal{D})$, where $\mathcal{D}\subset {\mathbb{Z}}^d[A]$ finite, that admit finite expansions of the form $\mathbf{x}={\mathbf{d}}_0+A{\mathbf{d}}_1+\cdots +A^{\ell -1}{\mathbf{d}}_{\ell -1}(\ell \in \mathbb{N},{\mathbf{d}}_0,\dots ,{\mathbf{d}}_{\ell -1}\in \mathcal{D})$for every element $\mathbf{x}\in {\mathbb{Z}}^d[A]$. We put special emphasis on the explicit computation of small digit sets $\mathcal{D}$ that admit this property for a given matrix A, using techniques from matrix theory, convex geometry, and the Smith Normal Form. Moreover, we provide a new proof of general results on this finiteness property and recover analogous finiteness results for digit systems in number fields a unified way.

让A成为$d\times d$具有没有特征值的有理元素的矩阵$\lambda \in \mathbb{C}$绝对值$\left|\lambda \right|<1$然后让${\mathbb{Z}}^d[\mathrm{一种}]$是最小的非平凡A -不变量$\mathbb{Z}$-模块。我们为构建数字系统制定了一个理论框架 $(\mathrm{一种},\mathcal{D})$， 在哪里$\mathcal{D}\subset {\mathbb{Z}}^d[\mathrm{一种}]$有限的，允许形式的有限扩展$\mathbf{X}={\mathbf{d}}_0+\mathrm{一种}{\mathbf{d}}_1+\cdots +{\mathrm{一种}}^{\ell -1}{\mathbf{d}}_{\ell -1}(\ell \in \mathbb{ñ},{\mathbf{d}}_0,\dots ,{\mathbf{d}}_{\ell -1}\in \mathcal{D})$对于每个元素$\mathbf{X}\in {\mathbb{Z}}^d[\mathrm{一种}]$. 我们特别强调小数字集的显式计算 $\mathcal{D}$使用矩阵理论、凸几何和史密斯范式中的技术，承认给定矩阵A的此属性。此外，我们提供了关于这种有限性性质的一般结果的新证明，并以统一的方式恢复了数字域中数字系统的类似有限性结果。