DISJOINT COVERING SYSTEMS OF RATIONAL BEATTY SEQUENCES

A rational Beatty sequence is a sequence {[alpha-n + beta]}, where alpha is rational, n runs through the integers and square brackets denote integer part. In 1973 Fraenkel conjectured that if {{[alpha(i)n + beta-1]}: i = 1,..., t} is a collection of rational Beatty sequences which partition the integers then alpha-i/alpha-j is an integer for some pair of distinct indices i and j. We show that the conjecture is true if alpha-i less-than-or-equal-to 2 for some i.