Topological properties of two-dimensional number systems

• Volume: 12, Issue: 1, page 69-79
• ISSN: 1246-7405

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Abstract

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In the two dimensional real vector space ${ℝ}^{2}$ one can define analogs of the well-known $q$-adic number systems. In these number systems a matrix $M$ plays the role of the base number $q$. In the present paper we study the so-called fundamental domain $ℱ$ of such number systems. This is the set of all elements of ${ℝ}^{2}$ having zero integer part in their “$M$-adic” representation. It was proved by Kátai and Környei, that $ℱ$ is a compact set and certain translates of it form a tiling of the ${ℝ}^{2}$. We construct points, where three different tiles of this tiling coincide. Furthermore, we prove the connectedness of $ℱ$ and give a result on the structure of its inner points.

How to cite

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Akiyama, Shigeki, and Thuswaldner, Jörg M.. "Topological properties of two-dimensional number systems." Journal de théorie des nombres de Bordeaux 12.1 (2000): 69-79. <http://eudml.org/doc/248483>.

@article{Akiyama2000,
abstract = {In the two dimensional real vector space $\mathbb \{R\}^2$ one can define analogs of the well-known $q$-adic number systems. In these number systems a matrix $M$ plays the role of the base number $q$. In the present paper we study the so-called fundamental domain $\mathcal \{F\}$ of such number systems. This is the set of all elements of $\mathbb \{R\}^2$ having zero integer part in their “$M$-adic” representation. It was proved by Kátai and Környei, that $\mathcal \{F\}$ is a compact set and certain translates of it form a tiling of the $\mathbb \{R\}^2$. We construct points, where three different tiles of this tiling coincide. Furthermore, we prove the connectedness of $\mathcal \{F\}$ and give a result on the structure of its inner points.},
author = {Akiyama, Shigeki, Thuswaldner, Jörg M.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {radix representation; fundamental domain; arcwise connected},
language = {eng},
number = {1},
pages = {69-79},
publisher = {Université Bordeaux I},
title = {Topological properties of two-dimensional number systems},
url = {http://eudml.org/doc/248483},
volume = {12},
year = {2000},
}

TY - JOUR
AU - Akiyama, Shigeki
AU - Thuswaldner, Jörg M.
TI - Topological properties of two-dimensional number systems
JO - Journal de théorie des nombres de Bordeaux
PY - 2000
PB - Université Bordeaux I
VL - 12
IS - 1
SP - 69
EP - 79
AB - In the two dimensional real vector space $\mathbb {R}^2$ one can define analogs of the well-known $q$-adic number systems. In these number systems a matrix $M$ plays the role of the base number $q$. In the present paper we study the so-called fundamental domain $\mathcal {F}$ of such number systems. This is the set of all elements of $\mathbb {R}^2$ having zero integer part in their “$M$-adic” representation. It was proved by Kátai and Környei, that $\mathcal {F}$ is a compact set and certain translates of it form a tiling of the $\mathbb {R}^2$. We construct points, where three different tiles of this tiling coincide. Furthermore, we prove the connectedness of $\mathcal {F}$ and give a result on the structure of its inner points.
LA - eng
KW - radix representation; fundamental domain; arcwise connected
UR - http://eudml.org/doc/248483
ER -

References

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1. [1] S. Akiyama, 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. Zbl0999.11065MR1738803
2. [2] S. Akiyama and T. Sadahiro, A self-similar tiling generated by the minimal pisot number. Acta Math. Info. Univ. Ostraviensis6 (1998), 9-26. Zbl1024.11066MR1822510
3. [3] W.J. Gilbert, Complex numbers with three radix representations. Can. J. Math.34 (1982), 1335-1348. Zbl0478.10007MR678674
4. [4] Complex bases and fractal similarity. Ann. sc. math. Quebec11 (1987), no. 1, 65-77. Zbl0633.10008MR912163
5. [5] M. Hata, On the structure of self-similar sets. Japan J. Appl. Math2 (1985), 381-414. Zbl0608.28003MR839336
6. [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. Zbl0765.54032MR1140724
7. [7] S. Ito, On the fractal curves induced from the complex radix expansion. Tokyo J. Math.12 (1989), no. 2, 299-320. Zbl0698.28002MR1030497
8. [8] I. Kátai, Number systems and fractal geometry. preprint. Zbl1029.11005
9. [9] I. Kátai and I. Környei, On number systems in algebraic number fields. Publ. Math. Debrecen41 (1992), no. 3-4, 289-294. Zbl0784.11049MR1189110
10. [10] I. Kátai and B. Kovács, Kanonische Zahlensysteme in der Theorie der Quadratischen Zahlen. Acta Sci. Math. (Szeged) 42 (1980), 99-107. Zbl0386.10007MR576942
11. [11] _Canonical number systems in imaginary quadratic fields. Acta Math. Hungar.37 (1981), 159-164. Zbl0477.10012MR616887
12. [12] I. Kátai and J. Szabó, Canonical number systems for complex integers. Acta Sci. Math. (Szeged) 37 (1975), 255-260. Zbl0309.12001MR389759
13. [13] D.E. Knuth, The art of computer programming, vol 2: Seminumerical algorithms, 3rd ed. Addison Wesley, London, 1998. Zbl0895.65001MR633878
14. [14] B. Kovács, Canonical number systems in algebraic number fields. Acta Math. Hungar.37 (1981), 405-407. Zbl0505.12001MR619892
15. [15] B. Kovács and A. Pethö, Number systems in integral domains, especially in orders of algebraic number fields. Acta Sci. Math. (Szeged) 55 (1991), 286-299. Zbl0760.11002MR1152592
16. [16] W. Müller, J.M. Thuswaldner, and R.F. Tichy, Fractal properties of number systems. Peri0dicaMathematica Hungarica, to appear. Zbl0980.11007
17. [17] J.M. Thuswaldner, Fractal dimension of sets induced by bases of imaginary quadratic fields, Math. Slovaca48 (1998), no. 4, 365-371. Zbl0956.11022MR1693521

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