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

TL;DR

This paper establishes an upper bound on the asymptotic repetitive threshold of balanced sequences using Fibonacci sequence colouring, revealing differences from the known repetitive threshold and suggesting new conjectures for even alphabet sizes.

Contribution

It introduces a novel Fibonacci-based colouring method to bound the asymptotic repetitive threshold of balanced sequences, highlighting key behavioral differences.

Findings

Bound is attained for d=2, 4, 8

Asymptotic threshold is at most 1+τ^3/2^{d-3}

Conjecture for infinitely many even d's

Abstract

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 $1+\frac{1}{d-2}$ for each $d \geq 3$. In contrast, our bound implies that the asymptotic repetitive threshold of $d$-ary balanced sequences is at most $1+\frac{\tau^3}{2^{d-3}}$ for each $d\geq 2$, where $\tau$ is the golden mean.