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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
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References:

[1] Akiyama, S., Self affine tiling and pisot numeration system. Number Theory and its Applications (K. Györy and S. Kanemitsu, eds.), Kluwer Academic Publishers, 1999, pp 7-17. · Zbl 0999.11065
[2] Akiyama, S. and Sadahiro, T., A self-similar tiling generated by the minimal pisot number. Acta Math. Info. Univ. Ostraviensis6 (1998), 9-26. · Zbl 1024.11066
[3] Gilbert, W.J., Complex numbers with three radix representations. Can. J. Math.34 (1982), 1335-1348. · Zbl 0478.10007
[4] Complex bases and fractal similarity. Ann. sc. math. Quebec11 (1987), no. 1, 65-77. · Zbl 0633.10008
[5] Hata, M., On the structure of self-similar sets. Japan J. Appl. Math2 (1985), 381-414. · Zbl 0608.28003
[6] Topological aspects of self-similar sets and singular functions. Fractal Geometry and Analysis (Netherlands) (J. Bélair and S. Dubuc, eds.), Kluwer Academic Publishers, 1991, pp. 255-276. · Zbl 0765.54032
[7] Ito, S., On the fractal curves induced from the complex radix expansion. Tokyo J. Math.12 (1989), no. 2, 299-320. · Zbl 0698.28002
[8] Kátai, I., Number systems and fractal geometry. preprint. · Zbl 1029.11005
[9] Kátai, I. and Környei, I., On number systems in algebraic number fields. Publ. Math. Debrecen41 (1992), no. 3-4, 289-294. · Zbl 0784.11049
[10] Kátai, I. and Kovács, B., Kanonische Zahlensysteme in der Theorie der Quadratischen Zahlen. Acta Sci. Math. (Szeged) 42 (1980), 99-107. · Zbl 0386.10007
[11] _Canonical number systems in imaginary quadratic fields. Acta Math. Hungar.37 (1981), 159-164. · Zbl 0477.10012
[12] Kátai, I. and Szabó, J., Canonical number systems for complex integers. Acta Sci. Math. (Szeged) 37 (1975), 255-260.
[13] Knuth, D.E., The art of computer programming, vol 2: Seminumerical algorithms, 3rd ed. Addison Wesley, London, 1998. · Zbl 0895.65001
[14] Kovács, B., Canonical number systems in algebraic number fields. Acta Math. Hungar.37 (1981), 405-407. · Zbl 0505.12001
[15] Kovács, B. and Pethö, A., Number systems in integral domains, especially in orders of algebraic number fields. Acta Sci. Math. (Szeged) 55 (1991), 286-299. · Zbl 0760.11002
[16] Müller, W., Thuswaldner, J.M., and Tichy, R.F., Fractal properties of number systems. Peri0dicaMathematica Hungarica, to appear. · Zbl 0980.11007
[17] Thuswaldner, J.M., Fractal dimension of sets induced by bases of imaginary quadratic fields, Math. Slovaca48 (1998), no. 4, 365-371. · Zbl 0956.11022
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