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 \(\mathcal F\).


11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
28A80 Fractals
11A63 Radix representation; digital problems
Full Text: DOI Numdam Numdam EuDML


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.