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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
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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.

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

References

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