Hostname: page-component-7dd5485656-c8zdz Total loading time: 0 Render date: 2025-10-31T22:59:13.427Z Has data issue: false hasContentIssue false
  • English
  • Français

The Bracket Function and Complementary Sets of Integers

Published online by Cambridge University Press:  20 November 2018

Aviezri S. Fraenkel
Aviezri S. Fraenkel*
Affiliation:
The Weizmann Institute of Science, Rehovot, Israel Bar-Ilan University, Ramat-Gan, Israel
Rights & Permissions [Opens in a new window]

Extract

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.

The following result is well known (as usual, [x]denotes the integral part of x):

(A) Let α and β be positive irrational numbers satisfying

1

Then the sets [nα], [nβ], n= 1, 2, …, are complementary with respect to the set of all positive integers]see, e.g. (1; 2; 4; 5; 6; 7; 8; 10; 13; 14; 15; 16). In some of these references the result, or a special case thereof, is mentioned in connection with Wythoff's game, with or without proof. It appears that Beatty (4) was the originator of the problem.

The theorem has a converse, and the following holds:

(B) Let α and β be positive. The sets [nα] and [nβ], n = 1, 2, …, are complementary with respect to the set of all positive integers if and only if α and β are irrational, and (1) holds.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 6 - 27
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Ahrens, W., Mathematische Unterhaltungen und Spiele, zweite vermehrte und verbesserte Auflage, Bd. I (Teubner, Leipzig, 1910).Google Scholar
2. Ball, W. W. R., Mathematical recreations and essays, revised by Coxeter, H. S. M., p. 39 (Macmillan, New York, 1947).Google Scholar
3. Bang, Th., On the sequence [na], n = 1, 2, …, Math. Scand. 5 (1957), 69–76.Google Scholar
4. Beatty, S., Problem 3173, Amer. Math. Monthly 33 (1926), 159; 34 (1927), 159.Google Scholar
5. Coxeter, H. S. M., The golden section, Phyllotaxis and Wythoff's game, Scripta Math. 19 (1953), 135–143.Google Scholar
6. Domoryad, A. P., Mathematical games and pastimes, translated by Moss, H. (Pergamon Press, Oxford, 1964).Google Scholar
7. Fan, Ky., The Dunkel memorial problem book, Problem 4399, p. 57.Google Scholar
8. Newman, D. J., Problem 5252, Amer. Math. Monthly 71 (1964), 1138; 72 (1965), 1144.Google Scholar
9. Niven, I., Diophantine approximations (Interscience, New York, 1963).Google Scholar
10. O'Beirne, T. H., Puzzles and paradoxes (Oxford Univ. Press, London, 1965).Google Scholar
11. Skolem, Th., On certain distributions of integers in pairs with given differences, Math. Scand. 5 (1957), 57–68.Google Scholar
12. Skolem, Th., Über einige Eigenschaften der Zahlenmengen [αn + /3] bei irrationalem a mit einleitenden Bemerkungen über einige kombinatorische Problème, Norske Vid. Selsk. Forh. (Trondheim) 30 (1957), 118–125.Google Scholar
13. Stewart, B. M., Theory of numbers, 2nd ed. (Macmillan, New York, 1964).Google Scholar
14. Uspensky, J. V. and Heaslet, M. A., Elementary number theory, p. 98 (McGraw-Hill, New York, 1939).Google Scholar
15. Wythoff, W. A., A modification of the game of Nim, Nieuw Arch. Wisk. 7 (1907), 199–202.Google Scholar
16. Yaglom, A. M. and Yaglom, I. M., Challenging mathematical problems with elementary solutions, translated by McCawley, J., Jr., revised and edited by Gordon, B., Vol. II (Holden-Day, San Francisco, 1967).Google Scholar