About the paper

coauthors Valérie Berthé
Anne Siegel
Wolfgang Steiner
Jörg M. Thuswaldner
language English
published in Advances in Mathematics 226 (2011)
pages 139 to 175
DOI 10.1016/j.aim.2010.06.010
supported by FWF, project S9610 (NFN S9600)


Shift radix systems form a collection of dynamical systems depending on a parameter r which varies in the d-dimensional real vector space. They generalise well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately related to fractal shapes, such as the classical Rauzy fractal and the twin dragon. These fractals turned out to be important for studying properties of expansions in several settings.
In the present paper we associate a collection of fractal tiles with shift radix systems. We show that for certain classes of parameters r these tiles coincide with affine copies of the well-known tiles associated with beta-expansions and canonical number systems. On the other hand, these tiles provide natural families of tiles for beta-expansions with (non-unit) Pisot numbers as well as canonical number systems with (nonmonic) expanding polynomials.
We also prove basic properties for tiles associated with shift radix systems. Indeed, we prove that under some algebraic conditions on the parameter r of the shift radix system, these tiles provide multiple tilings and even tilings of the d-dimensional real vector space. These tilings turn out to have a more complicated structure than the tilings arising from the known number systems mentioned above. Such a tiling may consist of tiles having infinitely many different shapes. Moreover, the tiles need not be self-affine (or graph directed self-affine).


  1. B. Adamczewski, C. Frougny, A. Siegel, W. Steiner, Rational numbers with purely periodic β-expansion, Bull. Lond. Math. Soc. 42 (2010), 538—552.
  2. S. Akiyama, Self affine tilings and Pisot numeration systems, K. Győory, S. Kanemitsu (Eds.), Number Theory, Its Applications, Kluwer, 1999, 1—17.
  3. S. Akiyama, On the boundary of self affine tilings generated by Pisot numbers, Math. Soc. Japan 54 (2002), 283—308.
  4. S. Akiyama, G. Barat, V. Berthé, A. Siegel, Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions, Monatsh. Math. 155 (2008), 377—419.
  5. S. Akiyama, T. Borbély, H. Brunotte, A. Pethő, J. M. Thuswaldner, Generalized radix representations and dynamical systems I, Acta Math. Hungar. 108 (2005), 207—238.
  6. S. Akiyama, H. Brunotte, A. Pethő, J. M. Thuswaldner, Generalized radix representations and dynamical systems II, Acta Arith. 121 (2006), 21—61.
  7. S. Akiyama, H. Brunotte, A. Pethő, J. M. Thuswaldner}, Generalized radix representations and dynamical systems III, Osaka J. Math. 45 (2008), 347—374.
  8. S. Akiyama, H. Brunotte, A. Pethő, J. M. Thuswaldner, Generalized radix representations and dynamical systems IV, Indag. Math. 19 (2008), 333—348.
  9. S. Akiyama, C. Frougny, J. Sakarovitch, Powers of rationals modulo 1 and rational base number systems, Israel J. Math. 168 (2008), 53—91.
  10. S. Akiyama, K. Scheicher, Intersecting two dimensional fractals and lines, Acta Sci. Math. (Szeged) 71 (2005), 555—580.
  11. S. Akiyama, J.M. Thuswaldner, Topological properties of two-dimensional number systems, J. Théor. Nombres Bordeaux 12 (2000), 69—79.
  12. S. Akiyama, J.M. Thuswaldner, A survey on topological properties of tiles related to number systems, Geom. Dedicata 109 (2004), 89—105.
  13. P. Arnoux, S. Ito, Pisot substitutions and Rauzy fractals, Bull. Belg. Math. Soc. Simon Stevin 8 (2001), 181—207, Journées Montoises d'Informatique Théorique, Marne-la-Vallée, 2000.
  14. G. Barat, V. Berthé, P. Liardet, J. Thuswaldner, Dynamical directions in numeration, Ann. Inst. Fourier (Grenoble) 56 (2006), 1987—2092.
  15. M. Barnsley, Fractals Everywhere, Academic Press Inc., Orlando, 1988.
  16. M. Barnsley, Superfractals, Cambridge University Press, 2006.
  17. V. Berthé, A. Siegel, Tilings associated with beta-numeration and substitutions, Integers 5 (2005), A2.
  18. N. Chekhova, P. Hubert, A. Messaoudi, Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci, J. Théor. Nombres Bordeaux 13 (2001), 371—394.
  19. K.J. Falconer, Fractal Geometry, John Wiley and Sons, Chichester, 1990.
  20. C. Frougny, B. Solomyak, Finite beta-expansions, Ergod. Th. and Dynam. Sys. 12 (1992), 713—723.
  21. M. Hollander, Linear numeration systems, finite beta expansions, and discrete spectrum of substitution dynamical systems, PhD thesis, Washington University, Seattle, 1996.
  22. P. Hubert, A. Messaoudi, Best simultaneous diophantine approximations of Pisot numbers and Rauzy fractals, Acta Arith. 124 (2006), 1—15.
  23. J.E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713—747.
  24. S. Ito, H. Rao, Atomic surfaces, tilings and coincidence. I. Irreducible case, Israel J. Math. 153 (2006), 129—155.
  25. C. Kalle, W. Steiner, Beta-expansions, natural extensions and multiple tilings associated with Pisot units, Trans. Amer. Math. Soc. 364 (2012), 2281—2318.
  26. I. Kátai, I. Kőrnyei, On number systems in algebraic number fields, Publ. Math. Debrecen 41 (1992), 289—294.
  27. D.E. Knuth, The Art of Computer Programming, vol. 2: Seminumerical Algorithms, 3rd ed., Addison-Wesley, London, 1998.
  28. B. Kovács, A. Pethő, Number systems in integral domains, especially in orders of algebraic number fields, Acta Sci. Math. (Szeged) 55 (1991), 286—299.
  29. K. Kuratowski, Topology, vol. I, Academic Press, Polish Scientific Publishers, New York, London, Warsaw, 1966.
  30. J. Lagarias, Y. Wang, Self-affine tiles in Rn, Adv. Math. 121 (1996), 21—49.
  31. J. Lagarias, Y. Wang, Integral self-affine tiles in Rn. II. Lattice tilings, J. Fourier Anal. Appl. 3 (1997), 83—102.
  32. R.D. Mauldin, S.C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988), 811—829.
  33. W. Parry, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401—416.
  34. A. Pethő, On a polynomial transformation and its application to the construction of a public key cryptosystem, in: Computational Number Theory, Debrecen, 1989, de Gruyter, Berlin, 1991, 31—43.
  35. G. Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France 110 (1982), 147—178.
  36. A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., 8 (1957), 477—493.
  37. K. Scheicher, P. Surer, J.M. Thuswaldner, C. van de Woestijne, Digit systems over commutative rings, Int. J. Number Theory 10 (2014), 14591483.
  38. K. Scheicher, J.M. Thuswaldner, Canonical number systems, counting automata and fractals, Math. Proc. Cambridge Philos. Soc. 133 (2002), 163—182.
  39. A. Siegel, Représentation des systèmes dynamiques substitutifs non unimodulaires, Ergodic Theory Dynam. Systems 23 (2003), 1247—1273.
  40. P. Surer, Characterisation results for shift radix systems, Math. Pannon., 18 (2007), 265—297.
  41. W.P. Thurston, Groups, tilings and finite state automata, Summer AMS Colloquium Lectures, 1989.
  42. A. Vince, Digit tiling of Euclidean space, in: Directions in Mathematical Quasicrystals, in: CRM Monogr. Ser., 13, Amer. Math. Soc., Providence, RI, 2000, 329—370.
  43. Y. Wang, Self-affine tiles, Advances in Wavelets, Hong Kong, 1997, Springer, Singapore, 1999, 261—282.


Advances in Mathematics
Austrian Science Foundation (FWF)
National Research Network (NFN) S9600