Consecutive Power Occurrences in Sturmian Words

TL;DR

This work extends Rampersad's bound on the ending-gap of cubes from the Fibonacci word to all Sturmian words, proving that the gap between consecutive ending positions of cubes is bounded by $10$ and that this bound is optimal, with the maximum realized by the Sturmian word of slope $\sqrt{2}-1$ and intercept $0$. It further shows a general phenomenon: for every $e\in\left[1,\frac{5+\sqrt{5}}{2}\right)$ there exists a bound $N=N(e)$ (depending only on $e$) such that the ending positions of $e$-powers with bounded period have gaps at most $N$. The key methods combine finite-bounded-period analysis (via balanced words and computational verification using Walnut) with a general combinatorial lemma about subword complexity, yielding a uniform-gap result for cubes and a parametric bound for $e$-powers. The paper also provides data for several other exponents, illustrating how the gap bounds behave as $e$ approaches the critical threshold, and highlighting the role of Ostrowski/Pell-based numeration in the analysis.

Abstract

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 $N$.

Theorems & Definitions (9)

• Theorem 1
• Theorem 2
• Lemma 3
• proof
• proof : Proof of Theorem 1
• Remark 4
• Lemma 5
• proof
• proof : Proof of Theorem \ref{['thm:genexp']}