We colour the Fibonacci sequence by suitable constant gap sequences to
provide an upper bound on the asymptotic repetitive threshold of d-ary
balanced sequences. The bound is attained for d=2,4 and 8 and we
conjecture that it happens for infinitely many even d's.
Our bound reveals an essential difference in behavior of the repetitive
threshold and the asymptotic repetitive threshold of balanced sequences. The
repetitive threshold of d-ary balanced sequences is known to be at least
1+d−21 for each d≥3. In contrast, our bound implies that the
asymptotic repetitive threshold of d-ary balanced sequences is at most
1+2d−3τ3 for each d≥2, where τ is the golden mean.Comment: arXiv admin note: text overlap with arXiv:2112.0285