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Descriptions of the Characteristic Sequence of an Irrational

Published online by Cambridge University Press:  20 November 2018

Tom C. Brown
Tom C. Brown*
Affiliation:
Department of Mathematics and Statistics Simon Fraser University Burnaby, British Columbia V5A 1S6
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Abstract

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Let α be a positive irrational real number. (Without loss of generality assume 0 < α < 1.) The characteristic sequence of α is

f(α) =f1f2 ···, where fn = [(n + 1)α] - [nα].

We make some observations on the various descriptions of the characteristic sequence of α which have appeared in the literature. We then refine one of these descriptions in order to obtain a very simple derivation of an arithmetic expression for [nα] which appears in A. S. Fraenkel, J. Levitt, and M. Shimshoni [17]. Some concluding remarks give conditions on n which are equivalent to fn = 1.

Keywords

10L10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 36 , Issue 1 , 01 March 1993 , pp. 15 - 21
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Anderson, P. G., personal communication.Google Scholar
2. Bernoulli, J. III, Sur une nouvelle espèce de calcul, Recueil pour les Astronomes, 1 Berlin, 1772,255–284.Google Scholar
3. Borel, J.-R and Laubie, F., Construction de mots de Christojfel, C. R. Acad. Sci. Paris (I) 313( 1991), 483– 485.Google Scholar
4. Borel, J.-R, Quelques mots sur la droite projective réelle, Séminaire de Théorie des Nombres de Bordeaux, to appear.Google Scholar
5. Brown, T. C., A characterization of the quadratic irrationals, Canad. Math. Bull. 34(1991 ), 36—41.Google Scholar
6. Brown, T. C., Sums of fractional parts of integer multiples of an irrational, preprint.Google Scholar
7. Cohn, H., Some direct limits of primitive homotopy words and of Markoff geodesies.In: Discontinuous Groups and Riemann Surfaces, Ann. of Math. Studies 79, Princeton University Press, Princeton, 1974, 81–98.Google Scholar
8. Connell, Ian G., Some properties of Beatty sequences II, Canad. Math. Bull. 3(1960), 17–22.Google Scholar
9. Crisp, D., Moran, W., Pollington, A. and Shiue, P., Substitution invariant cutting sequences, preprint.Google Scholar
10. Christoffel, E. B., Observatio Arithmetica, Annali di Matematica 6(1875), 145–152.Google Scholar
11. Fowler, D. H., Ratio in early Greek mathematics, Bull. AMS New Series 1(1979), 807–846.Google Scholar
12. Fowler, D. H., Book II of Euclid's Elements and a pre-Eudoxan theory of ratio, Archive for Hist, of Exact Sci. 22(1980), 5–56.Google Scholar
13. Fowler, D. H., Anthyphairetic ratio and Eudoxan proportion, Archive for Hist, of Exact Sci. 24(1981), 69–72.Google Scholar
14. Fraenkel, A. S., Systems of numeration, Amer. Math. Monthly 92(1985), 105–114.Google Scholar
15. Fraenkel, A. S., The use and usefulness of numeration systems, Information and Computation 81(1989), 46–61.Google Scholar
16. Fraenkel, A. S. and Borosh, I., A generalization of Whythoff's game, J. Combin. Theory, (A) 15(1973), 175–191.Google Scholar
17. Fraenkel, A. S., Levitt, J. and Shimshoni, M., Characterization of the set of values of f(n) = [nα], n = 1,2,…, Discrete Math. 2(1972), 335–345.Google Scholar
18. Fraenkel, A. S., Mushkin, M. and Tassa, U., Determination of [nθ] by its sequence of differences, Canad. Math. Bull. 21(1978), 441–446.Google Scholar
19. Hedlund, G. A., Sturmian minimal sets, Amer. J. Math. 66(1954), 605–620.Google Scholar
20. Ito, S. and Yasutomi, S., On continued fractions, substitutions and characteristic sequences [nx + y] — [(/i - l)x + y], Japan J. Math. 16(1990), 287–306.Google Scholar
21. Knorr, W. R., The Evolution of the Euclidean Elements, Reidel, Dordrecht, 1975.Google Scholar
22. Laubie, F., Prolongements homo graphique s de substitutions de mots de Christoffel, C. R. Acad. Sci. Paris (1)313(1991), 565–567.Google Scholar
23. Lekkerkerker, C. G., Representation of natural numbers as a sum of Fibonacci numbers, Simon Stevin 29(1952), 190–195.Google Scholar
24. Markoff, A. A., Sur une question de Jean Bernoulli, Math. Ann. 19(1882), 27–36.Google Scholar
25. Mignosi, F., Infinite words with linear subword complexity, Theor. Comput. Sci. 65(1989), 221–242.Google Scholar
26. Morse, M. and Hedlund, G. A., Symbolic dynamics II. Sturmian trajectories, Amer. J. Math. 62(1940), 1–42.Google Scholar
27. Nishioka, K., Shiokawa, I. and Tamura, J., Arithmetical properties of a certain power series, J. Number Theory, to appear.Google Scholar
28. Ostrowski, A., Bemerkungen zur Théorie der Diophantischen Approximation, Abh. Math. Sem. Hamburg 1(1922), 77–98.Google Scholar
29. Porta, H. and Stolarsky, K. B., Half-silvered mirrors and Wythoff's game, Canad. Math. Bull. 33(1990), 119–125.Google Scholar
30. Riddell, R. C., Eudoxan mathematics and the Eudoxan spheres, Archive for Hist, of Exact. Sci. 20(1979), 1–19.Google Scholar
31. Rosenblatt, J., The sequence of greatest integers of an arithmetic progression, J. London Math. Soc. (2) 17(1978), 213–218.Google Scholar
32. Series, C., The geometry of Markoff numbers, Math. Intelligencer 7(1985), 20–29.Google Scholar
33. Shallit, J., A generalization of automatic sequences, Theor. Comp. Sci. 61(1988), 1–16.Google Scholar
34. Shallit, J., Characteristic words as fixed points of homomorphisms, Univ. of Waterloo, Dept. of Computer Science, Tech. Report CS-91-72, 1991.Google Scholar
35. Smith, H. J. S., Note on continued fractions, Messenger of Math. 6(1876), 1–14.Google Scholar
36. Stolarsky, K. B., Beatty sequences, continued fractions, and certain shift operators, Canad. Math. Bull. 19(1976), 473–482.Google Scholar
37. Venkov, B. A., Elementary Number Theory, Translated and edited by Helen Alderson, Wolters-Noordhoff, Groningen, 1970,65–68.Google Scholar
38. Yaglom, A. M. and Yaglom, I. M., Challenging mathematical problems with elementary solutions, (trans. McCawley, J., Jr.), (revis, and ed. B. Gordon) 2, Holden-Day, San Francisco, 1967.Google Scholar
39. Zeckendorff, E., Representation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Royale Sci. Liège 42(1972), 179–182.Google Scholar
40. Zeeman, E. C., An algorithm for Eudoxan and anthipairetic ratios, preprint.Google Scholar