An upper bound on asymptotic repetition threshold of balanced sequences via colouring of the Fibonacci sequence

Abstract

We colour the Fibonacci sequence by suitable constant gap sequences to provide an upper bound on the asymptotic repetition 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 behaviour of the repetition threshold and the asymptotic repetition threshold of balanced sequences. The repetition threshold of d-ary balanced sequences is known to be at least 1+1d−2 for each d≥3. In contrast, our bound implies that the asymptotic repetition threshold of d-ary balanced sequences is at most 1+τ32d−3 for each d≥2, where τ is the golden mean.