Skip to main content
SpringerLink
Log in

Integral self-affine tiles in ℝn part II: Lattice tilings

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

Let A be an expanding n×n integer matrix with |det(A)|=m. Astandard digit set D for A is any complete set of coset representatives forℤ n/A(ℤ n). Associated to a given D is a setT (A, D), which is the attractor of an affine iterated function system, satisfyingT=∪ d∈D (T+d). It is known thatT (A, D) tilesℝ n by some subset ofℤ n. This paper proves that every standard digit set D gives a setT (A, D) that tilesℝ n with a lattice tiling.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bandt, C. (1991). Self-similar sets 5. Integer matrices and fractal tilings ofR n.Proc. Amer. Math Soc. 112, 549–562.

    Article  MATH  MathSciNet  Google Scholar 

  2. Berger, M. A., and Wang, Y. (1992). Multidimensional two-scale dilation equations. InWavelets. A Tutorial in Theory and Applications C. K. Chui, ed., Academic Press, New York, 295–323.

    Google Scholar 

  3. Cassels, J. W. S. (1957). An Introduction to Diophatine Approximations.Cambridge University Press, Cambridge.

    Google Scholar 

  4. Cerveau, D., Conze, J. P., and Raugi, A. (1995). Ensembles invariants pour un opérateur de transfert dans ℝd. preprint.

  5. Cohen, A. (1990). Ondelettes, analyses multirésolutions et filtres miriors em quadrature.Ann. Inst. Poincaré 7, 439–459.

    MATH  Google Scholar 

  6. Conze, J. P., and Raugi, A. (1990). Fonctions harmoniques pour un operateur de transition et applications.Bull. Soc. Math., France 118, 273–310.

    MATH  MathSciNet  Google Scholar 

  7. Conze, J. P., Hervé, L., and Raugi, A. (1995). Pavages auto-affines, opérateur de transfert et critères de réseau dans ℝd. preprint.

  8. De Boor, C., and Höllig, K. (1991). Box-spline tilings.Amer. Math. Monthly 98, 793–802.

    Article  MATH  MathSciNet  Google Scholar 

  9. Falconer, K. J. (1985).The Geometry of Fractal Sets. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  10. Gröchenig, K. (1994). Orthogonality criteria for compactly supported scaling functions.Appl. Comp. Harmonic Anal. 1, 242–245.

    Article  MATH  Google Scholar 

  11. Gröchenig, K., and Haas, A. (1994). Self-similar lattice tilings.J. Fourier Anal. 1, 131–170.

    Article  MATH  Google Scholar 

  12. Gröchenig, K., and Madych, W. (1992). Multiresolution analysis, Haar bases, and self-similar tilings.IEEE Trans. Inform. Theory 38 (2), 558–568.

    Article  Google Scholar 

  13. Grünbaum, B. and Shephard, G. C. (1987).Tilings and Patterns. W. H. Freeman, New York.

    MATH  Google Scholar 

  14. Hacon, D., Saldanha, N. C., and Veerman, J. J. P. (1995). Self-similar tilings of ℝn.Experimental Math.

  15. Hutchinson, J. E. (1981). Fractals and self-similarity.Indiana Univ. Math. J. 30, 713–747.

    Article  MATH  MathSciNet  Google Scholar 

  16. Kenyon, R. (1992). Self-replicating tilings. In:Symbolic Dynamics and Applications, P. Walters, ed., Contemporary Math. Vol., 135 239–264.

    MathSciNet  Google Scholar 

  17. Lagarias, J. C., and Wang, Y. (1996). Self-affine tiles inR n.Adv. Math. 121, 21–49.

    Article  MATH  MathSciNet  Google Scholar 

  18. ——. (1996). Integral self-affine tiles inR n I. Standard and nonstandard digit sets.J. London Math. Soc. 54, 161–179.

    MATH  MathSciNet  Google Scholar 

  19. ——. (1996). Haar bases forL 2(ℝn) and algebraic number theory.J. Number Theory 57, 181–197.

    Article  MATH  MathSciNet  Google Scholar 

  20. Lawton, W. (1991). Necessary and sufficient conditions for constructing orthonormal wavelet bases.J. Math. Phys. 32, 57–61.

    Article  MATH  MathSciNet  Google Scholar 

  21. Lawton, W. and Resnikoff, H. L. (1991). Multidimensional wavelet bases. AWARE Inc.

  22. McMullen, P. (1980). Convex bodies which tile space by translation.Mathematika 27, 113–121.

    Article  MATH  MathSciNet  Google Scholar 

  23. Newman, M. (1972).Integral Matrices. Academic Press, New York.

    MATH  Google Scholar 

  24. Odlyzko, A. M. (1978). Nonnegative digit sets in positional number systems.Proc. London Math. Soc. 37, 213–229.

    Article  MATH  MathSciNet  Google Scholar 

  25. Schrijver, L. (1965).Theory of linear and integer programming. John Wiley & Sons, New York.

    Google Scholar 

  26. Strichartz, R. S. (1993). Wavelets and self-affine tilings.Constructive Approximation.9, 327–346.

    Article  MATH  MathSciNet  Google Scholar 

  27. Taussky, O. (1957). On matrix classes corresponding to an ideal and its inverse.Illinois J. Math. 1, 108–113.

    MATH  MathSciNet  Google Scholar 

  28. Thurston, W. P. (1989). Groups, tilings and finite state automata. AMS Colloquium Lecture Notes, American Mathematical Society, Providence, RI.

    Google Scholar 

  29. Venkov, B. A. (1954). On a class of Euclidean polyhedra.Vestnik Leningrad Univ. Ser. Mat. Fiz. Him. 9, 11–31 (Russian).

    MathSciNet  Google Scholar 

  30. Vince, A. (1993). Replicating tesselations.SIAM J. Discrete Math. 3 501–521.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. AT&T Labs-Research, 600 Mountain Avenue, 07974, Murray Hill, NJ

    Jeffrey C. Lagarias

  2. School of Mathematics, Georgia Institute of Technology, 30332, Atlanta, GA

    Yang Wang

Authors
  1. Jeffrey C. Lagarias
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lagarias, J.C., Wang, Y. Integral self-affine tiles in ℝn part II: Lattice tilings. The Journal of Fourier Analysis and Applications 3, 83–102 (1997). https://doi.org/10.1007/BF02647948

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02647948

Math Subject Classifications

Keywords and Phrases

Search

Navigation