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arXiv · 2211.11877

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

Lubomíra Dvořáková·Edita Pelantová
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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.

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Lubomíra Dvořáková, Edita Pelantová. 2022-11-21. An upper bound on asymptotic repetitive threshold of balanced sequences via colouring of the Fibonacci sequence. https://arxiv.org/abs/2211.11877

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