Geodesic
The Bracket Function and Complementary Sets of Integers
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 6-27

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Fraenkel, Aviezri S. The Bracket Function and Complementary Sets of Integers. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 6-27. doi: 10.4153/CJM-1969-002-7
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