Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T01:25:40.211Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterization of two-distance sequences

Part of: Sequences and sets Designs and configurations Discrete mathematics in relation to computer science

Published online by Cambridge University Press:  09 April 2009

W. F. Lunnon  and
P. A. B. Pleasants
W. F. Lunnon
Affiliation:
Department of Computer ScienceSt. Patrick's CollegeMaynoothCounty KildareEire
P. A. B. Pleasants
Affiliation:
School of Mathematics, Physics Computing and ElectronicsMacQuarie UniversityNSW 2109, Australia
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.

Three differently defined classes of two-symbol sequences, which we call the two-distance sequences, the linear sequences and the characteristic sequences, have been discussed by a number of authors and some equivalences between them are known. We present a self-contained proof that the three classes are the same (when ambiguous cases of linear sequences are suitably in terpreted). Associated with each sequence is a real invariant (having a different appropriate definition for each of the three classes). We give results on the relation between sequences with the same invariant and on the symmetry of the sequences. The sequences are closely related to Beatty sequences and occur as digitized straight lines and quasicrystals. They also provide examples of minimal word proliferation in formal languages.

Keywords

Beatty sequencesFibonacci sequencessymbolic dynamicschain codesdigitized straight linesquasicrystals.

MSC classification

Secondary: 11B99: None of the above, but in this section 05B99: None of the above, but in this section 68R05: Combinatorics 68U05: Computer graphics; computational geometry
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 53 , Issue 2 , October 1992 , pp. 198 - 218
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Berger, M. A., Felzenbaum, A. and Fraenkel, A. S., ‘Disjoint covering systems of rational Beatty sequences’, J. Combin. Theory Ser. A 42 (1986), 150–153.CrossRefGoogle Scholar
[2]Boshernitzan, M. and Fraenkel, A. S., ‘Nonhomogeneous spectra of numbers’, Discrete Math. 34 (1981), 325–327.CrossRefGoogle Scholar
[3]Boshernitzan, M. and Fraenkel, A. S., ‘A linear algorithm for nonhomogeneous spectra of numbers’, J. Algorithms 5 (1984), 187–198.CrossRefGoogle Scholar
[4]Bruckstein, A. M., ‘The self-similarity of digital straight lines’, Proceedings of the 10th International Conference on Pattern Recognition, Atlantic City, 1990.Google Scholar
[5]Bruckstein, A. M., ‘Self-similarity properties of digitized straight lines’, Technical Report no. 616, Dept. of Computer Science, Technion, Israel Institute of Technology, 1990.Google Scholar
[6]Coven, E. M. and Hedlund, G. A., ‘Sequences with minimal block growth’, Math. Systems Theory 7 (1973), 138–153.CrossRefGoogle Scholar
[7]Cusick, T. W. and Flahive, M. E., The Markoff and Lagrange spectra (Mathematical surveys and monographs no. 30, Amer. Math. Soc., Providence, Rhode Island, 1989).CrossRefGoogle Scholar
[8]Graham, R. L., ‘Covering the positive integers by sets of the form {[nαα + β}: n = 1, 2, …}’, J. Combin. Theory Ser. A 15 (1973), 354–358.CrossRefGoogle Scholar
[9]Grünbaum, B. and Shephard, G. C., Tilings and Patterns (Freeman, San Francisco, 1986).Google Scholar
[10]Hardy, G. H. and Wright, E. M., ‘An introduction to the theory of numbers’, (4th ed., Clarendon Press, Oxford, 1960).Google Scholar
[11]Lunnon, W. F. and Pleasants, P. A. B., ‘Quasicrystallographic tilings’, J. Math. Pures Appl. 66 (1987), 217–263.Google Scholar
[12]Markoff, A., ‘Sur les formes binaires indéfinies’, Math. Ann. 15 (1879), 381–406.CrossRefGoogle Scholar
[13]Markoff, A., ‘Sur les formes binaires indéfinies’, Math. Ann. 17 (1880), 379–400.CrossRefGoogle Scholar
[14]Markoff, A., ‘Sur une question de Jean Bernoulli’, Math. Ann. 19 (1882), 27–36.CrossRefGoogle Scholar
[15]Morse, M. and Hedlund, G. A., ‘Symbolic dynamics’, Amer. J. Math. 60 (1938), 815–866.CrossRefGoogle Scholar
[16]Morse, M. and Hedlund, G. A., ‘Symbolic dynamics II. Sturmian trajectories’, Amer. J. Math. 62 (1940), 1–42.CrossRefGoogle Scholar
[17]Pleasants, P. A. B., ‘Quasicrystailography: some interesting new patterns’ in Elementary and analytic the theory of numbers, (Banach Center Publications, Vol. 17, PWN-Polish Scientific Publishers, Warsaw, 1985), 439–461.Google Scholar
[18]Series, C., ‘The geometry of Markoff numbers’, Math. Intelligencer 7 (3) (1985), 20–29.CrossRefGoogle Scholar
[19]Simpson, R. J., ‘Disjoint covering systems of rational Beatty sequences’, Discrete Math. (to appear).Google Scholar
[20]Stolarsky, K. B., ‘Beatty sequences, continued fractions and certain shift operators’, Canad. Math. Bull. 19 (1976), 473–482.CrossRefGoogle Scholar
[21]Mignosi, F., ‘On the number of factors of Sturmian words’, Theoretical Computer Science 82 (1991), 71–84.CrossRefGoogle Scholar
[22]Séébold, P., ‘Fibonacci morphism and Sturmian words’, Theoretical Computer Science 88 (1991), 365–384.CrossRefGoogle Scholar