Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-22T21:35:59.305Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Existence of Similar Sublattices

Published online by Cambridge University Press:  20 November 2018

J. H. Conway ,
E. M. Rains  and
N. J. A. Sloane
J. H. Conway
Affiliation:
Mathematics Department, Princeton University, Princeton, NJ 08540, USA
E. M. Rains
Affiliation:
Information Sciences Research, AT&T Shannon Lab, Florham Park, NJ 07932-0971, USA
N. J. A. Sloane
Affiliation:
Information Sciences Research, AT&T Shannon Lab, Florham Park, NJ 07932-0971, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Partial answers are given to two questions. When does a lattice $\Lambda $ contain a sublattice ${\Lambda }'$ of index $N$ that is geometrically similar to $\Lambda $? When is the sublattice “clean”, in the sense that the boundaries of the Voronoi cells for ${\Lambda }'$ do not intersect $\Lambda $?

Keywords

52C07
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 6 , 01 December 1999 , pp. 1300 - 1306
Copyright
Copyright © Canadian Mathematical Society 1999

References

[1] Baake, M., Solution of the coincidence problem in dimensions d ≤ 4. In: The Mathematics of Long-Range Aperiodic Order (ed. Moody, R. V.), Kluwer, Dordrecht, 1997, 199–237.Google Scholar
[2] Baake, M. and Moody, R. V., Similarity submodules and semigroups. In: Quasicrystals and Discrete Geometry (ed. Patera, J.), Fields InstituteMonographs 10, Amer.Math. Soc., Providence, RI, 1998, 1–13.Google Scholar
[3] Baake, M. and Moody, R. V., Similarity Submodules and Root Systems in Four Dimensions. Canad. J. Math. 51(1999), 1258–1276.Google Scholar
[4] Baake, M. and Pleasants, P. A. B., Algebraic solution of the coincidence problem in two and three dimensions. Z. Naturforschung 50A(1995), 711–717.Google Scholar
[5] Baake, M. and Pleasants, P. A. B., The coincidence problem for crystals and quasicrystals. In: Aperiodic ‘94 (eds. Chapuis, G. and Paciorek, W.), World Scientific, Singapore, 1995, 25–29.Google Scholar
[6] Chapman, R. J., Shrinking integer lattices. Discrete Math. 142(1995), 39–48.Google Scholar
[7] Chapman, R. J., Shrinking integer lattices II. J. Pure Appl. Alg. 78(1992), 123–129.Google Scholar
[8] Conway, J. H. and Sloane, N. J. A., Low-dimensional lattices I: Quadratic forms of small determinant. Proc. Royal Soc. London 418A(1988), 17–41.Google Scholar
[9] Conway, J. H. and Sloane, N. J. A., Low-dimensional lattices VI: Voronoi reduction of three-dimensional lattices. Proc. Royal Soc. London 436A(1992), 55–68.Google Scholar
[10] Conway, J. H. and Sloane, N. J. A., Sphere Packings, Lattices and Groups. 3rd edn, Springer-Verlag, NY, 1998.Google Scholar
[11] Kitaoka, Y., Arithmetic of Quadratic Forms. Cambridge University Press, Cambridge, 1993.Google Scholar
[12] O’Meara, O. T., Introduction to Quadratic Forms. Springer-Verlag, NY, 1971.Google Scholar
[13] Pleasants, P. A. B., Baake, M. and Roth, J., Planar coincidences for N-fold symmetry. J. Math. Phys. 37(1996), 1029–1058.Google Scholar
[14] Ramanujan, S., On the expression of a number in the form ax2 + by2 + cz2 + u2. Proc. Camb. Philos. Soc. 19(1917), 11–21; Collected Papers, Cambridge University Press, Cambridge, 1927, 169–178.Google Scholar
[15] Servetto, S. D., Vaishampayan, V. A. and Sloane, N. J. A., Multiple description lattice vector quantization. In: Proceedings DCC ‘99: Data Compression Conference (Snowbird, 1999) (eds. Storer, J. A. and M. Cohn), IEEE Computer Society, Los Alamitos, CA, 1999, 13–22.Google Scholar
[16] Sloane, N. J. A., The On-Line Encyclopedia of Integer Sequences. Published electronically at http://www.research.att.com/njas/sequences/.Google Scholar
[17] Vaishampayan, V. A., Sloane, N. J. A. and Servetto, S. D., Multiple description vector quantization with lattice codebooks: design and analysis. In preparation.Google Scholar
[18] Watson, G. L., Integral Quadratic Forms. Cambridge University Press, Cambridge, 1960.Google Scholar