The Bracket Function and Complementary Sets of Integers

Source abstract

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.