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Balancedness constants for m-bonacci sequences

Determine, for each integer m ≥ 2 and each k ≥ 1, the minimal integer C^{(m)}_k such that the m-bonacci sequence x_m, defined as the fixed point of the substitution 0→01, 1→02, …, m−2→0(m−1), m−1→0, is (k, C^{(m)}_k)-balanced.

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Background

The paper proves precise balancedness results for Fibonacci (m=2) and Tribonacci (m=3) words, showing uniform (k,2)-balancedness across all k for Tribonacci and improved bounds for Fibonacci. These results motivate extending balancedness characterization to the entire family of m-bonacci sequences.

Only partial results are known for the case k=1, as noted by prior work; the general determination for all k remains open.

References

Given in~\cref{thm:Fib k2balanced,thm:Trib k2balanced} on the Fibonacci and Tribonacci sequences, we raise the following question. For an integer $m\ge 2$, let ${x}_m$ be the $m$-bonacci sequence, fixed point of $0\mapsto 01, 1 \mapsto 02, \ldots, m-2 \mapsto 0(m-1), m-1 \mapsto 0$. What is the value of the smallest integer $C{(m)}_k\ge 1$ such that ${x}_m$ is $(k,C{(m)}_k)$-balanced? Bounds on $C{(m)}_1$ are given in.

— Effective Computation of Generalized Abelian Complexity for Pisot Type Substitutive Sequences (2504.13584 - Couvreur et al., 18 Apr 2025) in Section 7 (Open problems and questions)