The Bracket Function and Complementary Sets of Integers

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 satisfying1Then 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.