Complementing and Exactly Covering Sequences

Abstract The following is proved (in a slightly more general setting): Let α1, …, αm be positive real, γ1, …, γm real, and suppose that the system [nαi + γi], i = 1, …, m, n = 1, 2, …, contains every positive integer exactly once (= a complementing system). Then α i α j is an integer for some i ≠ j in each of the following cases: (i) m = 3 and m = 4; (ii) m = 5 if all αi but one are integers; (iii) m ⩾ 5, two of the αi are integers, at least one of them prime; (iv) m ⩾ 5 and αn ⩽ 2n for n = 1, 2, …, m − 4. For proving (iv), a method of reduction is developed which, given a complementing system of m sequences, leads under certain conditions to a derived complementing system of m − 1 sequences.