Partitioning the positive integers to seven Beatty sequences

AbstractA conjecture of Fraenkel asserts that any partition of the positive integers into m ⪰ 3 sets {⌊αin + βi⌊}n=1∞ - Beatty sequences - with real constants ai and βi, and α1 > α2 > … > αm > 1 satisfies α=2m − 12i − 1,1 ≤ i ≤ m. Fraenkel's conjecture was proved using balanced words by Tijdeman for m = 3, by Altman, Gaujal and Hordijk for m = 4 and by Tijdeman for m = 5, 6. We use an approach similar to the last one to settle the conjecture for seven sequences