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

2022/11/21 by Ľubomíra Dvořáková, Edita Pelantová, Dvořáková, Lubomíra +1
Computer Science · Engineering · #68R15 #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2211.11877

openalex publication_date 2022/11/21 · openalex created_date 2022/11/29 · openalex updated_date 2026/07/28

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+(1)/(d-2) for each d ≥ 3. In contrast, our bound implies that the asymptotic repetitive threshold of d-ary balanced sequences is at most 1+\fracτ³2^d-3 for each d≥ 2, where τ is the golden mean.

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• Asymptotic repetitive threshold of balanced sequences
• The limit of repetition thresholds of rich sequences
• The asymptotic repetition threshold of sequences rich in palindromes
• The repetition threshold of episturmian sequences
• Generalised Fibonacci sequences constructed from balancing words