Net Occurrences in Fibonacci and Thue-Morse Words

TL;DR

This work advances the study of net occurrences by resolving two long-standing conjectures: every Fibonacci word contains exactly three net occurrences and every Thue-Morse word contains exactly nine. The authors introduce overlapping net occurrence cover (ONOC) to dramatically prune the search space, reducing the problem to analyzing a small set of bridging sub-occurrences. They also develop precise recurrence characterizations for occurrences of smaller-order Fibonacci and Thue-Morse words, yielding structural insight and enabling counting of small-order nets. The results deepen the combinatorial understanding of these canonical words and provide tools with potential impact on algorithmic analyses and the study of net occurrences in morphic sequences.

Abstract

A net occurrence of a repeated string in a text is an occurrence with unique left and right extensions, and the net frequency of the string is the number of its net occurrences in the text. Originally introduced for applications in Natural Language Processing, net frequency has recently gained attention for its algorithmic aspects. Guo et al. [CPM 2024] and Ohlebusch et al. [SPIRE 2024] focus on its computation in the offline setting, while Guo et al. [SPIRE 2024], Inenaga [arXiv 2024], and Mieno and Inenaga [CPM 2025] tackle the online counterpart. Mieno and Inenaga also characterize net occurrences in terms of the minimal unique substrings of the text. Additionally, Guo et al. [CPM 2024] initiate the study of net occurrences in Fibonacci words to establish a lower bound on the asymptotic running time of algorithms. Although there has been notable progress in algorithmic developments and some initial combinatorial insights, the combinatorial aspects of net occurrences have yet to be thoroughly examined. In this work, we make two key contributions. First, we confirm the conjecture that each Fibonacci word contains exactly three net occurrences. Second, we show that each Thue-Morse word contains exactly nine net occurrences. To achieve these results, we introduce the notion of overlapping net occurrence cover, which narrows down the candidate net occurrences in any text. Furthermore, we provide a precise characterization of occurrences of Fibonacci and Thue-Morse words of smaller order, offering structural insights that may have independent interest and potential applications in algorithm analysis and combinatorial properties of these words.

Figures (8)

• Figure 1: An example for \ref{['def:onoc']}. The set $\{(1,6),(4,9),(9,14)\}$ is an ONOC, with each of its net occurrences underlined in blue; $\{(4,6),(9,9)\}$ is the corresponding set of BNSOs. Note that $(2, 7)$ is a net occurrence outside of this ONOC, underlined in orange.
• Figure 2: An illustration of \ref{['thm:fib-occ-inductive']} when $j = 4$. Each row depicts a factorization of $F_i$ with relevant factors highlighted in colors. The top two, middle two, and bottom two rows correspond to sets $\Theta_{i,j-2}$, $\Theta_{i,j-1}$ and $\Theta_{i,j}$, respectively. Each green and blue occurrence of $F_{i-j}$ is introduced by an occurrence of $F_{i-(j-2)}$ and $F_{i-(j-1)}$, respectively. The yellow occurrence is the rightmost one.
• Figure 3: An illustration of the occurrences of $\mathcal{T}\xspace_{i-j}$ and $\overline{\mathcal{T}\xspace_{i-j}}\xspace$ in $\mathcal{T}\xspace_{i}$ for $1 \leq j \leq 6$.
• Figure 4: An illustration of \ref{['thm:tm-occ']}. Each dark blue, pink, and light blue occurrence of $\mathcal{T}\xspace_{i-j}$ is introduced by an occurrence of $\mathcal{T}\xspace_{i-(j-1)}$, $\overline{\mathcal{T}\xspace_{i-(j-1)}}\xspace$, and $\mathcal{T}\xspace_{i-(j-2)}$ respectively. Each occurrence of $\mathcal{T}\xspace_{i-j}$ that is both dark blue and pink indicates that it is introduced by both an occurrence of $\mathcal{T}\xspace_{i-(j-1)}$ and an occurrence of $\overline{\mathcal{T}\xspace_{i-(j-1)}}\xspace$.
• Figure 5: An illustration of several factorizations of $F_i$ from \ref{['thm:f-i-fac']} and \ref{['thm:q-i-delta']} where $\Delta := \Delta(1 - (i \bmod 2))$ and $\Delta' := \Delta(i \bmod 2)$. Net occurrences of $F_{i-2}$ and $F_{i-2} \ Q_i$ are in yellow and green, respectively. Super-occurrences of the two BNSOs are shown as arrows.

Theorems & Definitions (46)

• Definition 1: Net occurrence conf/cpm/2024/guo
• Lemma 2: journal/ipl/1995/droubay
• Lemma 3: journal/tit/2021/navarro
• Lemma 5: Overlap-free book/1997/lothaire
• Lemma 6: Cube-free book/1997/lothaire
• Remark 8
• Definition 9: ONOC and BNSO
• Lemma 9
• Lemma 10: journal/tit/2021/navarro
• Lemma 12: journal/tit/2021/navarro