## Topological properties of two-dimensional number systems.(English)Zbl 1012.11072

Let $$A,B$$ be integers such that $$-1\leq A \leq B$$ with $$B \geq 2$$, and assume that the polynomial $$X^2 + AX + B$$ is irreducible. Let $${\mathcal N}=\{(0,0)^T,\dots,(B-1,0)^T \}$$ and $$M=\left( \begin{matrix} 0 & -B \cr 1 & -A \end{matrix} \right)$$.
The fundamental domain of the number system $$(M,{\mathcal N})$$ is defined by ${\mathcal F} = \{z : z = \sum_{j \geq 1}M^{-j}d_j, d_j\in {\mathcal N}\}.$ A point $$z \in {\mathcal F}$$ is called a vertex if there exist nonzero distinct integer vectors $$g_1,g_2 \in \mathbb Z^2$$ such that $$z \in ({\mathcal F}+g_1)\cup ({\mathcal F}+g_2)$$. It is proved in this paper that if $$A \not= 0$$ then $$\mathcal F$$ has at least six vertices. If $$2A=B+3$$, then $$\mathcal F$$ has infinitely many, and if $$2A>B+3$$, then $$\mathcal F$$ has uncountably many vertices.
Concerning the topological properties it is proved that $$\mathcal F$$ is arcwise connected and each point with a finite $$M$$-adic representation is an inner point of $$\mathcal F$$.

### MSC:

 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension 28A80 Fractals 11A63 Radix representation; digital problems

### Keywords:

radix representation; fundamental domain; arcwise connected
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### References:

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