Topological properties of two-dimensional number systems
Journal de théorie des nombres de Bordeaux, Volume 12 (2000) no. 1, pp. 69-79.

In the two dimensional real vector space 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 of such number systems. This is the set of all elements of 2 having zero integer part in their “M-adic” representation. It was proved by Kátai and Környei, that is a compact set and certain translates of it form a tiling of the 2 . We construct points, where three different tiles of this tiling coincide. Furthermore, we prove the connectedness of and give a result on the structure of its inner points.

Pour une matrice réelle M d’ordre 2 donnée, on peut définir la notion de représentation M-adique d’un élément de 2 . On note le domaine fondamental constitué des nombres de 2 dont le développement “M-adique” ne commence pas par 0. C’est l’analogue dans 2 des nombres q-adiques, où la matrice M joue le rôle de la base q. Kátai et Környei ont démontré que est compact, et que 2 s’écrit comme la réunion dénombrable de certains translatés de , l’intersection de 2 quelconques d’entre eux étant de mesure nulle. Dans cet article, nous construisons des points qui appartiennent simultanément à trois translatés de , et nous montrons que est connexe. Nous donnons aussi une propriété sur la structure des points intérieurs de .

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Akiyama, Shigeki; Thuswaldner, Jörg M. Topological properties of two-dimensional number systems. Journal de théorie des nombres de Bordeaux, Volume 12 (2000) no. 1, pp. 69-79. http://www.numdam.org/item/JTNB_2000__12_1_69_0/

[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. | MR | Zbl

[2] S. Akiyama and T. Sadahiro, A self-similar tiling generated by the minimal pisot number. Acta Math. Info. Univ. Ostraviensis 6 (1998), 9-26. | EuDML | MR | Zbl

[3] W.J. Gilbert, Complex numbers with three radix representations. Can. J. Math. 34 (1982), 1335-1348. | MR | Zbl

[4] Complex bases and fractal similarity. Ann. sc. math. Quebec 11 (1987), no. 1, 65-77. | MR

[5] M. Hata, On the structure of self-similar sets. Japan J. Appl. Math 2 (1985), 381-414. | MR | Zbl

[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. | MR

[7] S. Ito, On the fractal curves induced from the complex radix expansion. Tokyo J. Math. 12 (1989), no. 2, 299-320. | MR | Zbl

[8] I. Kátai, Number systems and fractal geometry. preprint.

[9] I. Kátai and I. Környei, On number systems in algebraic number fields. Publ. Math. Debrecen 41 (1992), no. 3-4, 289-294. | MR | Zbl

[10] I. Kátai and B. Kovács, Kanonische Zahlensysteme in der Theorie der Quadratischen Zahlen. Acta Sci. Math. (Szeged) 42 (1980), 99-107. | MR | Zbl

[11] _Canonical number systems in imaginary quadratic fields. Acta Math. Hungar. 37 (1981), 159-164. | MR | Zbl

[12] I. Kátai and J. Szabó, Canonical number systems for complex integers. Acta Sci. Math. (Szeged) 37 (1975), 255-260. | MR | Zbl

[13] D.E. Knuth, The art of computer programming, vol 2: Seminumerical algorithms, 3rd ed. Addison Wesley, London, 1998. | MR | Zbl

[14] B. Kovács, Canonical number systems in algebraic number fields. Acta Math. Hungar. 37 (1981), 405-407. | MR | Zbl

[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. | MR | Zbl

[16] W. Müller, J.M. Thuswaldner, and R.F. Tichy, Fractal properties of number systems. Peri0dica Mathematica Hungarica, to appear. | Zbl

[17] J.M. Thuswaldner, Fractal dimension of sets induced by bases of imaginary quadratic fields, Math. Slovaca 48 (1998), no. 4, 365-371. | MR | Zbl