Consecutive Power Occurrences in Sturmian Words
<|reference_start|>Consecutive Power Occurrences in Sturmian Words: We show that every Sturmian word has the property that the distance between consecutive ending positions of cubes occurring in the word is always bounded by $10$ and this bound is optimal, extending a result of Rampersad, who proved that the bound $9$ holds for the Fibonacci word. We then give a general result showing that for every $e \in [1,(5+\sqrt{5})/2)$ there is a natural number $N$, depending only on $e$, such that every Sturmian word has the property that the distance between consecutive ending positions of $e$-powers occurring in the word is uniformly bounded by $N$.<|reference_end|>
@article{bell2024consecutive,
    title={Consecutive Power Occurrences in Sturmian Words},
    author={Jason Bell, Chris Schulz, Jeffrey Shallit},
    journal={arXiv preprint arXiv:2402.09597},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.09597},
    primaryClass={math.CO cs.DM cs.FL math.NT}
}