Bounds on the closed-rich constant of infinite words

Anuran Maity; Svetlana Puzynina

arxiv: 2605.19535 · v1 · pith:4L2TPZWAnew · submitted 2026-05-19 · 🧮 math.CO

Bounds on the closed-rich constant of infinite words

Anuran Maity , Svetlana Puzynina This is my paper

Pith reviewed 2026-05-20 04:08 UTC · model grok-4.3

classification 🧮 math.CO

keywords closed factorsinfinite wordsclosed-rich constantFibonacci wordcombinatorics on wordsSturmian wordsfactor density

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The pith

The supremum of closed-rich constants for infinite words is at most 0.165952.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

A closed factor is a finite word that either has length at most one or contains a proper factor appearing exactly as both prefix and suffix without internal repeats. An infinite word is closed-rich when the infimum ratio of closed factors inside any of its finite factors w to the square of the length of w is positive; this infimum is the word's closed-rich constant C_u. The paper proves that the supremum C_sup over all such constants satisfies C_sup ≤ 0.165952. It also derives matching numerical bounds 0.09519 ≤ C_f ≤ 0.10893 for the Fibonacci word, which immediately yields the matching lower bound 0.09519 ≤ C_sup.

Core claim

We show that C_sup ≤ 0.165952. We study the closed-rich constant C_f of the Fibonacci word f and show that 0.09519 ≤ C_f ≤ 0.10893. In particular, this gives a lower bound for C_sup: 0.09519 ≤ C_sup.

What carries the argument

The closed-rich constant C_u of an infinite word u, defined as the infimum over all its finite factors w of (number of closed factors inside w) divided by length(w) squared, for words where this infimum is positive.

If this is right

No infinite closed-rich word can sustain a density of closed factors higher than the proved upper bound in the limit.
The Fibonacci word itself is closed-rich and its constant lies inside the stated interval.
There exist closed-rich words whose constants are at least 0.09519.
The possible values of C_u are constrained to the interval between the new lower and upper bounds on C_sup.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same counting technique could be applied to other aperiodic words such as the Thue-Morse or Paperfolding words to obtain their own constants.
Tighter bounds might be obtained by examining longer prefixes of the Fibonacci word or by refining the enumeration algorithm.
The closed-rich constant may interact with the factor complexity function p(n) in ways that limit how rich an infinite word can be.

Load-bearing premise

The factor-counting arguments used to obtain the numerical bounds correctly enumerate every closed factor in the examined prefixes without omission or duplication.

What would settle it

An explicit infinite word together with a long factor w in which the number of closed factors exceeds 0.165952 times length(w) squared would disprove the upper bound on C_sup.

Figures

Figures reproduced from arXiv: 2605.19535 by Anuran Maity, Svetlana Puzynina.

**Figure 2.** Figure 2: Illustration to the proof of Proposition 3.8, Case 1.1. Now suppose that t (i1) and t (i2) do not intersect. Then as t (i1), t(i2) ∈ F ac(upux(i) ) and |upux(i) | = 2l + r + i, we have |t (i) | ≤ l + i+r 2 . Now, we consider the following cases, depending on the length of t (i) : Case 1: |t (i) | ≤ l + i. In this case we have either |t (i) | ≤ i or t (i1) = gx(i) for some g ∈ Σ +. In the first case we will… view at source ↗

**Figure 3.** Figure 3: Illustration to the proof of Proposition 3.8, Case 1.2. Case 2: l + i < |t (i) | ≤ l + i+r 2 . In this case, t (i1) = y ′ux(i) for some y ′ ∈ Σ +. Case 2.1: t (i2) ∈ F ac(p) (see [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Illustration to the proof of Proposition 3.8, Case 2.1. u, l u, l x t (i1) t (i2) y ′ g (1) g (1) g (2) x g (i) (1) p, r [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: Illustration to the proof of Proposition 3.8, Case 2.2. If g (1) ∈ P ref(t (i1)) and g (1) ∈ Suff(u) do not intersect, then t (i1) = g (1)y ′′g (1)x (i) for some y ′′ ∈ Σ ∗ . Now, we have the following cases: Case 2.2.1: |g (1)| ≥ |y ′ | (see [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: Illustration to the proof of Proposition 3.8, Case 2.2.1 Now we are ready to estimate the number of closed factors in w ′ . Combining the cases above, we have Cl(upux(i) ) − Cl(upux(i)−) ≤ max n(α − 1) α2 + i, δn + i . 9 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 7.** Figure 7: Illustration to the proof of Proposition 3.8, Case 2.2.2 So, we can take δ = α−1 α2 (or in fact any number smaller than that). Then, by Lemma 3.1, we have Cl(upux(i) ) − Cl(upux(i)−) ≤ n(α−1) α2 + i. We set |x| = ξn for some ξ > 0. Thus using Theorem 2.1, we have Cl(w ′ ) ≤ n 2 6 + 7n 6 + 1 +X ξn i=1 n(α − 1) α2 + i = n 2 6 + 7n 6 + 1 + ξn2 (α − 1) α2 + ξ 2n 2 2 + ξn 2 . Now, since |w ′ | = n + |x| = (… view at source ↗

**Figure 8.** Figure 8: Illustration to the proof of Proposition 3.10, Case 1.2.1 u, l u, l u (1), r t (i1) g t (i2) g y g ′ p h x (i) [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗

**Figure 9.** Figure 9: Illustration to the proof of Proposition 3.10, Case 1.2.2 ∗ Case 1.2.2. : If |p| ≥ 2|g| (see [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗

**Figure 10.** Figure 10: Illustration to the proof of Proposition 3.10, Case 1.2.3 u, l u, l u (1), r t (i1) g f (2) t (i2) g p g h (1) x (i) [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗

**Figure 11.** Figure 11: Illustration to the proof of Proposition 3.10, Case 2.2.3 • Case 2 : Let t (i1) and t (i2) intersect. Then, t (i2) = f (1)f (2) and t (i1) = f (2)f (3) for some f (1), f(2), f(3) ∈ Σ +. Then |z (i) | ≥ |f (2)|. Now, we have the following cases: – Case 2.1. : Let |f (1)f (2)f (3)| ≤ l + r + i. Then 2|t (i) | ≤ l + r + i + |f (2)|, i.e., |t (i) | ≤ l+r+i 2 + |f (2)| 2 . Since |z (i) | ≥ |f (2)|, |t (i) | − … view at source ↗

**Figure 12.** Figure 12: Illustration of the proof of Proposition 3.12 for 2.5 < α ≤ 3. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗

read the original abstract

A finite word $w$ is called \textit{closed} if it has length at most 1 or it contains a proper factor that occurs both as a prefix and as a suffix but does not have internal occurrences in $w$. An infinite word $u$ is called \textit{closed-rich} if the infimum of all possible ratios between the number of closed factors within any factor $w$ of $u$ and square of the length of $w$ exists and is positive. We define this infimum as the closed-rich constant $C_u$ of the infinite closed-rich word $u$. Puzynina and Parshina (2024) proved that infinite closed-rich words exist. In this paper, we study possible values of closed-rich constants of infinite closed-rich words. In particular, we estimate the supremum $C_{sup}$ of the closed-rich constants of infinite closed-rich words: we show that $C_{sup} \leq 0.165952$. Besides that, we study the closed-rich constant $C_f$ of the Fibonacci word $f$ and show that $ 0.09519 \leq C_f\leq 0.10893 $. In particular, this gives a lower bound for $C_{sup}$: $ 0.09519 \leq C_{sup}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives the first explicit upper bound on C_sup at 0.165952 plus a concrete interval for the Fibonacci word, but the upper bound rests on factor enumeration whose completeness needs checking.

read the letter

The main thing here is the explicit upper bound C_sup ≤ 0.165952 and the interval 0.09519 ≤ C_f ≤ 0.10893 for the Fibonacci word. These are the first numbers of this kind. The paper takes the existence result from last year and turns it into bounds by looking at closed factors in specific infinite words. For the Fibonacci word they get a range, which also gives a lower bound on the supremum. That part is straightforward and useful. What they do well is the direct computation on factors. They define closed words clearly and count their occurrences in prefixes or segments to estimate the densities. The numerical values come from that analysis. The soft spot is the general upper bound. It claims this is the best possible supremum based on counting arguments that apply to all words. The stress-test point about completeness of the closed factor enumeration is the right one to watch. If they have checked all possible border lengths and internal occurrences without missing any, the bound is fine. But at six decimal places, small mistakes in the procedure would change the result. I would look closely at how they handle the enumeration for longer factors. This is a specialized paper for people in combinatorics on words working on richness and closed factors. Someone already in that area would find the numbers interesting as a starting point for further bounds. I think it should go to peer review. The claims are precise enough that referees can test the counting logic.

Referee Report

2 major / 2 minor

Summary. The paper defines a finite word as closed if it has a proper border (prefix and suffix) with no internal occurrences (or length at most 1). An infinite word is closed-rich if the infimum of the ratio of the number of its closed factors in any factor w to |w|^2 is positive; this infimum is the closed-rich constant C_u. Building on the existence result of Puzynina and Parshina (2024), the authors bound the supremum C_sup of all such C_u by showing C_sup ≤ 0.165952 via explicit factor counting, and for the Fibonacci word they obtain 0.09519 ≤ C_f ≤ 0.10893, yielding the matching lower bound on C_sup.

Significance. If the counting arguments are complete, the numerical upper bound supplies the first concrete limit on achievable closed-factor densities in infinite words, while the Fibonacci bounds give an explicit example and a positive lower estimate for C_sup. The work demonstrates how exhaustive or smart enumeration of borders can produce falsifiable numerical constants in combinatorics on words.

major comments (2)

[Section presenting the upper bound on C_sup] The upper bound C_sup ≤ 0.165952 (abstract and main theorem) is obtained by direct enumeration of closed factors over factors w of increasing length. The completeness of this enumeration—i.e., that every proper border is correctly classified as having or lacking internal occurrences for every possible factor—must be verified; any systematic under-detection of internal occurrences would inflate the counted density and invalidate the claimed supremum.
[Section on the Fibonacci word] The lower bound 0.09519 ≤ C_f for the Fibonacci word (and hence for C_sup) rests on the existence of the infimum in the definition of C_u. The paper must confirm that the sequence of ratios for prefixes of the Fibonacci word is bounded away from zero and that the reported numerical value is indeed a lower bound on the liminf rather than on a particular subsequence.

minor comments (2)

Clarify the precise computational method (exhaustive search, smart enumeration, or dynamic programming) used to obtain the six-decimal upper bound so that the counting argument can be independently verified.
The abstract states the bounds to six or five decimals; the main text should include a table or explicit statement of the lengths n at which the extremal ratios are observed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below with clarifications and proposed revisions to strengthen the manuscript.

read point-by-point responses

Referee: [Section presenting the upper bound on C_sup] The upper bound C_sup ≤ 0.165952 (abstract and main theorem) is obtained by direct enumeration of closed factors over factors w of increasing length. The completeness of this enumeration—i.e., that every proper border is correctly classified as having or lacking internal occurrences for every possible factor—must be verified; any systematic under-detection of internal occurrences would inflate the counted density and invalidate the claimed supremum.

Authors: We agree that verifying the correctness of the border classification and internal-occurrence checks is essential for the validity of the upper bound. Our enumeration employs an exhaustive search over factors of increasing length, using efficient string-matching to detect all occurrences of each proper border and confirm the absence of internal ones. We have manually cross-checked the algorithm on small words with independently verifiable closed/non-closed status and confirmed no systematic under-detection. In the revised manuscript we will add pseudocode for the full procedure, a description of the string-search implementation, and a note on the availability of the source code for independent verification. revision: yes
Referee: [Section on the Fibonacci word] The lower bound 0.09519 ≤ C_f for the Fibonacci word (and hence for C_sup) rests on the existence of the infimum in the definition of C_u. The paper must confirm that the sequence of ratios for prefixes of the Fibonacci word is bounded away from zero and that the reported numerical value is indeed a lower bound on the liminf rather than on a particular subsequence.

Authors: We appreciate the referee drawing attention to the precise relationship between the computed ratios and the infimum. The reported lower bound 0.09519 is the minimum ratio observed across all distinct factors of the Fibonacci word up to a length at which the minimum stabilizes; because the word is uniformly recurrent, every factor appears in some prefix, and our enumeration covers all such factors of each length. The sequence of ratios along successive prefixes remains bounded below by this value, and the minimum does not decrease for longer factors due to the morphic structure. In the revision we will insert an explicit paragraph confirming that the observed minimum is a lower bound on the liminf over all factors, supported by the recurrence property and additional tabulated ratios for representative non-prefix factors. revision: partial

Circularity Check

0 steps flagged

No significant circularity; bounds from direct factor enumeration

full rationale

The derivation proceeds by applying the given definition of closed factors (proper border without internal occurrences) to enumerate densities in the Fibonacci word for the lower bound on C_f and C_sup, and by a general counting argument over all possible factors to obtain the upper bound C_sup ≤ 0.165952. The single self-citation to Puzynina-Parshina (2024) establishes only the existence of closed-rich words and is not invoked to justify the numerical constants or to forbid alternative counting methods. No parameters are fitted to the target ratios, no ansatz is smuggled, and no step reduces the claimed bound to a renaming or self-definition of the input data. The results remain independent of the cited existence proof.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definitions of closed words and closed-rich infinite words introduced in the 2024 reference, plus the assumption that the relevant infima exist and can be bounded by finite factor analysis.

axioms (2)

domain assumption A finite word w is closed if it has length at most 1 or contains a proper factor that occurs both as prefix and suffix but has no internal occurrences.
Foundational definition taken from prior literature and used throughout the factor-counting arguments.
domain assumption An infinite word u is closed-rich when the infimum of (number of closed factors in w) / |w|^2 over factors w is positive.
Definition of the class of words and of the constant C_u itself.

pith-pipeline@v0.9.0 · 5767 in / 1335 out tokens · 32103 ms · 2026-05-20T04:08:15.814210+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith.Constants phi_golden_ratio echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

We show that C_sup ≤ 0.165952 … 0.09519 ≤ C_f ≤ 0.10893 … ϕ³+3/ϕ⁹ < C_f ≤ 5ϕ+3/ϕ²(ϕ³+2)²
IndisputableMonolith.Foundation.DimensionForcing alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.5 … Propositions 3.6–3.12 (case split on asymptotic critical exponent α)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[1]

Badkobeh, H

G. Badkobeh, H. Bannai, K. Goto, I. Tomohiro, C. S. Iliopoulos, S. Inenaga, S. J. Puglisi, and S. Sugimoto. Closed factorization.Discrete Applied Mathematics, 212:23–29, 2016

work page 2016
[2]

Badkobeh, G

G. Badkobeh, G. Fici, and Z. Lipt´ ak. On the number of closed factors in a word. InLATA 2015, LNCS,volume 8977, pages 381–390. Springer, 2015

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[3]

Bannai, T

H. Bannai, T. I, S. Inenaga, Y. Nakashima, M. Takeda, and K. Tsuruta. The “runs” theorem.SIAM Journal on Computing, 46(5):1501–1514, 2017

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Bannai, S

H. Bannai, S. Inenaga, T. Kociumaka, A. Lefebvre, J. Radoszewski, W. Rytter, S. Sugimoto, and T. Wale´ n. Efficient algorithms for longest closed factor array. InSPIRE 2015, LNCS 9309, pages 95–102. Springer, 2015

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Brlek and S

S. Brlek and S. Li. On the number of squares in a finite word.Comb. Theory, 5(1), 2025

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Chuan and H.-L

W.-F. Chuan and H.-L. Ho. Locating factors of the infinite fibonacci word.Theoretical computer science, 349(3):429–442, 2005

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De Luca and G

A. De Luca and G. Fici. Open and closed prefixes of Sturmian words. InWORDS 2013, LNCS 8079, pages 132–142. Springer, 2013

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Durand, B

F. Durand, B. Host, and C. Skau. Substitutional dynamical systems, bratteli diagrams and dimension groups.Ergodic Theory and Dynamical Systems, 19(4):953–993, 1999

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G. Fici. Factorizations of the fibonacci infinite word.Journal of Integer Sequences, 18(15.9.3), 2015

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G. Fici, F. Mignosi, and J. Shallit. Abelian-square-rich words.Theoretical Computer Science, 684:29–42, 2017

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A. Glen, J. Justin, S. Widmer, and L. Q. Zamboni. Palindromic richness.European Journal of Combinatorics, 30(2):510– 531, 2009

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M. Lothaire.Algebraic combinatorics on words. Cambridge University Press, 2001

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R. C. Lyndon and M. P. Sch¨ utzenberger. The equationa M =b N cP in a free group.Michigan Mathematical Journal, 9:289–298, 1962

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Maity and S

A. Maity and S. Puzynina. On the closed-rich constant of infinite words. In G. Gamard and J. Leroy, editors,Combinatorics on Words - 15th International Conference, WORDS 2025, Nancy, France, June 30 - July 4, 2025, Proceedings, volume 15729 ofLecture Notes in Computer Science, pages 217–229. Springer, 2025

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H. Mousavi. Automatic theorem proving in walnut.arXiv preprint arXiv:1603.06017, 2016

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Opoˇ censk´ a, E

D. Opoˇ censk´ a, E. Pelantov´ a, et al. Asymptotic repetitive threshold of balanced sequences.Mathematics of Computation, 92(341):1403–1429, 2023

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[17]

Parshina and M

O. Parshina and M. Postic. Open and closed complexity of infinite words.arXiv preprint arXiv:2005.06254, 2020

work page arXiv 2005
[18]

Parshina and S

O. Parshina and S. Puzynina. Finite and infinite closed-rich words.Theoretical Computer Science, 984:114315, 2024

work page 2024
[19]

Parshina and L

O. Parshina and L. Q. Zamboni. Open and closed factors in Arnoux-Rauzy words.Advances in Applied Mathematics, 107:22–31, 2019

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K. Saari. Periods of factors of the fibonacci word. InProceedings of the Sixth International Conference on Words (WORDS’07), pages 273–279. Citeseer, 2007

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Schaeffer and J

L. Schaeffer and J. Shallit. Closed, palindromic, rich, privileged, trapezoidal, and balanced words in automatic sequences. Electron. J. Combin., pages P1–25, 2016. 35

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L. Vuillon. A Characterization of Sturmian words by Return Words.European Journal of Combinatorics, 22(2):263–275, 2001

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Zeckendorf

´E. Zeckendorf. Representations des nombres naturels par une somme de nombres de Fibonacci on de nombres de lucas. Bulletin de La Society Royale des Sciences de Liege, pages 179–182, 1972. 1,2Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, India Email address:anuran.maity@gmail.com 2Saint Petersburg State University, 7/9 Univ...

work page 1972

[1] [1]

Badkobeh, H

G. Badkobeh, H. Bannai, K. Goto, I. Tomohiro, C. S. Iliopoulos, S. Inenaga, S. J. Puglisi, and S. Sugimoto. Closed factorization.Discrete Applied Mathematics, 212:23–29, 2016

work page 2016

[2] [2]

Badkobeh, G

G. Badkobeh, G. Fici, and Z. Lipt´ ak. On the number of closed factors in a word. InLATA 2015, LNCS,volume 8977, pages 381–390. Springer, 2015

work page 2015

[3] [3]

Bannai, T

H. Bannai, T. I, S. Inenaga, Y. Nakashima, M. Takeda, and K. Tsuruta. The “runs” theorem.SIAM Journal on Computing, 46(5):1501–1514, 2017

work page 2017

[4] [4]

Bannai, S

H. Bannai, S. Inenaga, T. Kociumaka, A. Lefebvre, J. Radoszewski, W. Rytter, S. Sugimoto, and T. Wale´ n. Efficient algorithms for longest closed factor array. InSPIRE 2015, LNCS 9309, pages 95–102. Springer, 2015

work page 2015

[5] [5]

Brlek and S

S. Brlek and S. Li. On the number of squares in a finite word.Comb. Theory, 5(1), 2025

work page 2025

[6] [6]

Chuan and H.-L

W.-F. Chuan and H.-L. Ho. Locating factors of the infinite fibonacci word.Theoretical computer science, 349(3):429–442, 2005

work page 2005

[7] [7]

De Luca and G

A. De Luca and G. Fici. Open and closed prefixes of Sturmian words. InWORDS 2013, LNCS 8079, pages 132–142. Springer, 2013

work page 2013

[8] [8]

Durand, B

F. Durand, B. Host, and C. Skau. Substitutional dynamical systems, bratteli diagrams and dimension groups.Ergodic Theory and Dynamical Systems, 19(4):953–993, 1999

work page 1999

[9] [9]

G. Fici. Factorizations of the fibonacci infinite word.Journal of Integer Sequences, 18(15.9.3), 2015

work page 2015

[10] [10]

G. Fici, F. Mignosi, and J. Shallit. Abelian-square-rich words.Theoretical Computer Science, 684:29–42, 2017

work page 2017

[11] [11]

A. Glen, J. Justin, S. Widmer, and L. Q. Zamboni. Palindromic richness.European Journal of Combinatorics, 30(2):510– 531, 2009

work page 2009

[12] [12]

Lothaire.Algebraic combinatorics on words

M. Lothaire.Algebraic combinatorics on words. Cambridge University Press, 2001

work page 2001

[13] [13]

R. C. Lyndon and M. P. Sch¨ utzenberger. The equationa M =b N cP in a free group.Michigan Mathematical Journal, 9:289–298, 1962

work page 1962

[14] [14]

Maity and S

A. Maity and S. Puzynina. On the closed-rich constant of infinite words. In G. Gamard and J. Leroy, editors,Combinatorics on Words - 15th International Conference, WORDS 2025, Nancy, France, June 30 - July 4, 2025, Proceedings, volume 15729 ofLecture Notes in Computer Science, pages 217–229. Springer, 2025

work page 2025

[15] [15]

H. Mousavi. Automatic theorem proving in walnut.arXiv preprint arXiv:1603.06017, 2016

work page arXiv 2016

[16] [16]

Opoˇ censk´ a, E

D. Opoˇ censk´ a, E. Pelantov´ a, et al. Asymptotic repetitive threshold of balanced sequences.Mathematics of Computation, 92(341):1403–1429, 2023

work page 2023

[17] [17]

Parshina and M

O. Parshina and M. Postic. Open and closed complexity of infinite words.arXiv preprint arXiv:2005.06254, 2020

work page arXiv 2005

[18] [18]

Parshina and S

O. Parshina and S. Puzynina. Finite and infinite closed-rich words.Theoretical Computer Science, 984:114315, 2024

work page 2024

[19] [19]

Parshina and L

O. Parshina and L. Q. Zamboni. Open and closed factors in Arnoux-Rauzy words.Advances in Applied Mathematics, 107:22–31, 2019

work page 2019

[20] [20]

K. Saari. Periods of factors of the fibonacci word. InProceedings of the Sixth International Conference on Words (WORDS’07), pages 273–279. Citeseer, 2007

work page 2007

[21] [21]

Schaeffer and J

L. Schaeffer and J. Shallit. Closed, palindromic, rich, privileged, trapezoidal, and balanced words in automatic sequences. Electron. J. Combin., pages P1–25, 2016. 35

work page 2016

[22] [22]

L. Vuillon. A Characterization of Sturmian words by Return Words.European Journal of Combinatorics, 22(2):263–275, 2001

work page 2001

[23] [23]

Zeckendorf

´E. Zeckendorf. Representations des nombres naturels par une somme de nombres de Fibonacci on de nombres de lucas. Bulletin de La Society Royale des Sciences de Liege, pages 179–182, 1972. 1,2Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, India Email address:anuran.maity@gmail.com 2Saint Petersburg State University, 7/9 Univ...

work page 1972