Gaussian Behavior and Geometric Gaps in Decompositions from Recurrences with Zero Coefficients

Sajad Salami

arxiv: 2604.15447 · v1 · submitted 2026-04-16 · 🧮 math.NT

Gaussian Behavior and Geometric Gaps in Decompositions from Recurrences with Zero Coefficients

Sajad Salami This is my paper

Pith reviewed 2026-05-10 09:41 UTC · model grok-4.3

classification 🧮 math.NT

keywords Zeckendorf representationslinear recurrence sequencesGaussian distributiongeometric gapsgreedy decompositionsLagonacci sequencenon-unique representations

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The pith

Decompositions from recurrences with a zero leading coefficient still yield a Gaussian distribution of summands and geometrically decaying gaps.

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

The paper examines representations of positive integers as sums drawn from sequences obeying linear recurrences that begin with a zero coefficient, using the Lagonacci sequence as the main example. Although these sequences permit many valid decompositions, the greedy choice of largest possible terms produces a canonical form whose count of summands converges in distribution to a Gaussian. The distances between the positions of the selected terms follow a geometric decay law. These regularities extend to a broad family of similar recurrences through an equivalence principle for statistical ensembles. The paper also establishes that the total number of admissible decompositions grows exponentially at a precise rate of two.

Core claim

For Zero Linear Recurrence Relations with leading coefficient zero, the canonical greedy decomposition of positive integers has a number of summands that converges in distribution to a Gaussian, while the gaps between the selected indices decay geometrically. These statistical features remain stable across a wide class of such relations by the principle of equivalence of ensembles. The multiplicity of representations is quantified by showing that the number of valid decompositions increases exponentially at rate exactly 2.

What carries the argument

The canonical greedy decomposition for a Zero Linear Recurrence Relation (ZLRR), which selects the largest allowable terms while respecting the recurrence, and which carries the distribution analysis even after uniqueness is lost.

If this is right

The number of summands in the greedy decomposition converges in distribution to a Gaussian.
The gaps between selected indices in the decomposition follow a geometric distribution.
The Gaussian and geometric behaviors persist for a wide class of ZLRRs through the equivalence of ensembles.
The total number of legal decompositions grows exponentially with rate exactly 2.

Where Pith is reading between the lines

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

Distributional regularity may prove more durable than uniqueness in additive bases built from recurrences.
Analogous Gaussian and geometric features could appear in other representation systems that drop uniqueness while retaining a greedy selection rule.
The exact growth rate of 2 suggests an underlying binary branching structure in the allowed term choices.

Load-bearing premise

That a canonical greedy decomposition remains well-defined for these zero-coefficient recurrences and that the equivalence of ensembles principle transfers the desired statistical properties from the strictly positive coefficient setting.

What would settle it

Direct computation of greedy Lagonacci decompositions for a sequence of large integers whose summand counts fail to approach a normal distribution or whose index gaps deviate from geometric frequencies.

read the original abstract

Zeckendorf's theorem establishes a unique representation for positive integers as sums of non-consecutive Fibonacci numbers. This result has been generalized to Positive Linear Recurrence Sequences (PLRS), where key statistical properties, such as the Gaussian distribution of summands, depend on strictly positive recurrence coefficients. This paper investigates the consequences of relaxing this condition by studying \textit{Zero Linear Recurrence Relations (ZLRRs)}, where the leading coefficient is zero ($c_1=0$). Focusing on the \textit{Lagonacci sequence} ($Z_{n+1}=Z_{n-1}+Z_{n-2}$) as a primary case study, we demonstrate that while the uniqueness of decompositions is lost, fundamental statistical behaviors persist. We prove that the number of summands in the canonical greedy decomposition converges to a \textit{Gaussian distribution} and that the distribution of gaps between indices decays \textit{geometrically}. Furthermore, we utilize the \textit{principle of equivalence of ensembles} to show these properties are robust for a wide class of ZLRRs. Finally, we quantify the non-uniqueness of these systems, proving that the number of legal decompositions grows \textit{exponentially} at a rate $\alpha =2$, significantly exceeding the growth of the underlying sequence.

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

Zero-coefficient recurrences preserve Gaussian summand counts and geometric gaps via ensemble equivalence, but that step needs explicit mixing checks.

read the letter

This paper shows that dropping the leading coefficient to zero in linear recurrence sequences does not break the Gaussian distribution on the number of summands or the geometric decay of gaps in the greedy decompositions. For the Lagonacci sequence it also pins down that the total number of legal decompositions grows exponentially at rate exactly 2, which is larger than the growth of the sequence itself. The main new piece is the systematic treatment of the zero-leading-coefficient case, which had been left out of the earlier PLRS literature because uniqueness fails. The authors correctly note that the greedy rule still produces a well-defined decomposition and then track its statistics directly. They further claim the same behaviors hold for a wider family by invoking equivalence of ensembles. That extension is the part that could matter beyond the single example. The work is honest about the loss of uniqueness and gives a concrete quantitative replacement for it. The soft spot is the ensemble step. When the leading coefficient is zero the recurrence changes the dependence graph, so the usual transfer-matrix spectral gap or rapid-mixing arguments from the positive-coefficient setting do not apply automatically. The paper does not appear to supply an independent check that the microcanonical and canonical measures still coincide asymptotically or that the required ergodicity holds. Without that, the claim that the Gaussian and geometric properties are robust for a wide class rests on an unverified analogy. For readers working on statistical properties of additive bases or Zeckendorf-type representations this is a natural next case to examine. It is not a large shift in the field, but it fills a clear gap with specific, checkable statements. I would bring it to a reading group to walk through the ensemble justification. I would not cite it in my own work in the next year. It deserves peer review because the setting is well-posed and the claims are narrow enough that referees can test them directly.

Referee Report

2 major / 2 minor

Summary. The paper extends Zeckendorf decompositions to Zero Linear Recurrence Sequences (ZLRRs) with leading coefficient c1=0, taking the Lagonacci sequence Z_{n+1}=Z_{n-1}+Z_{n-2} as the main example. It claims to prove that the number of summands in the canonical greedy decomposition converges in distribution to a Gaussian, that gaps between indices in these decompositions decay geometrically, that these behaviors are robust across a wide class of ZLRRs by the principle of equivalence of ensembles, and that the total number of legal decompositions grows exponentially at rate α=2.

Significance. If the proofs are complete and the ensemble equivalence is rigorously justified, the work would show that Gaussian summand counts and geometric gap statistics survive the loss of uniqueness that occurs when the leading recurrence coefficient vanishes. The explicit exponential rate for the multiplicity of representations supplies a quantitative measure of non-uniqueness that is absent from the classical positive-coefficient theory.

major comments (2)

[§4] §4 (Equivalence of ensembles for ZLRRs): The manuscript invokes the principle of equivalence of ensembles to transfer Gaussian and geometric-gap results from strictly positive PLRS to ZLRRs with c1=0, yet supplies no verification that the required spectral gap or rapid mixing persists after the linear dependence graph is altered by the recurrence Z_{n+1}=Z_{n-1}+Z_{n-2}. Without an explicit check that the microcanonical and canonical measures coincide asymptotically, the robustness claim for the “wide class” rests on an unverified analogy.
[Theorem 5.1] Theorem 5.1 (exponential growth rate α=2): The proof that the number of legal decompositions grows at rate exactly 2 is stated without an explicit computation of the growth constant (e.g., via the largest eigenvalue of the associated transfer matrix or generating-function radius) or a comparison to the growth rate of the underlying sequence itself; the assertion that this rate “significantly exceeds” the sequence growth is therefore unsupported by the displayed bounds.

minor comments (2)

[Abstract] The abstract asserts that “proofs exist” for Gaussian convergence and geometric decay but the main text does not indicate where the error bounds or convergence rates are derived; adding forward references to the relevant propositions would improve readability.
[§2] Notation for the Lagonacci sequence is introduced without an explicit initial-condition table; a short table of the first ten terms would clarify the greedy decomposition examples used later.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments highlight areas where additional explicit verifications will strengthen the manuscript. We address each major comment below and will incorporate the suggested clarifications in a revised version.

read point-by-point responses

Referee: [§4] §4 (Equivalence of ensembles for ZLRRs): The manuscript invokes the principle of equivalence of ensembles to transfer Gaussian and geometric-gap results from strictly positive PLRS to ZLRRs with c1=0, yet supplies no verification that the required spectral gap or rapid mixing persists after the linear dependence graph is altered by the recurrence Z_{n+1}=Z_{n-1}+Z_{n-2}. Without an explicit check that the microcanonical and canonical measures coincide asymptotically, the robustness claim for the “wide class” rests on an unverified analogy.

Authors: We agree that an explicit verification of the spectral gap is needed to make the application of ensemble equivalence fully rigorous. In the revised manuscript we will add to §4 a direct computation of the transition matrix for the Lagonacci recurrence (and for the general ZLRR class under the stated coefficient conditions). We will show that the matrix is irreducible and aperiodic with second-largest eigenvalue strictly less than 1 in modulus, confirming a uniform spectral gap independent of n. This establishes rapid mixing and the asymptotic coincidence of microcanonical and canonical measures, thereby justifying the transfer of Gaussian and geometric-gap statistics to the zero-coefficient setting. revision: yes
Referee: [Theorem 5.1] Theorem 5.1 (exponential growth rate α=2): The proof that the number of legal decompositions grows at rate exactly 2 is stated without an explicit computation of the growth constant (e.g., via the largest eigenvalue of the associated transfer matrix or generating-function radius) or a comparison to the growth rate of the underlying sequence itself; the assertion that this rate “significantly exceeds” the sequence growth is therefore unsupported by the displayed bounds.

Authors: The current proof sketch of Theorem 5.1 uses a transfer-matrix count of admissible index sequences but does not display the matrix or its characteristic polynomial. We will revise the theorem and its proof to include the explicit 2×2 (or small) transfer matrix whose largest eigenvalue is exactly 2, together with the radius of the associated generating function. We will also compute the growth rate of the Lagonacci sequence itself (the unique real root of x³−x−1=0, approximately 1.3247) and state the strict inequality 2 > ρ_Z explicitly. These additions will make the claimed exponential rate and the comparison fully supported by displayed bounds. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations rest on independent recurrence theory and ensemble equivalence

full rationale

The paper's core results—the Gaussian limit for the number of summands in the greedy decomposition of the Lagonacci sequence, the geometric decay of gaps, and the exponential growth rate α=2 for the number of legal decompositions—are presented as direct proofs for the specific recurrence Z_{n+1}=Z_{n-1}+Z_{n-2}. The extension to a wider class of ZLRRs is justified by invoking the principle of equivalence of ensembles, a standard tool from statistical mechanics whose applicability is argued separately rather than derived from the target statistics themselves. No equation reduces a claimed prediction to a fitted parameter by construction, no uniqueness theorem is imported solely via self-citation, and no ansatz is smuggled in; the loss of uniqueness is explicitly acknowledged and quantified independently. The derivation chain therefore remains self-contained against external benchmarks from Zeckendorf theory and linear recurrences.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; paper assumes standard results from Zeckendorf theory and applies equivalence of ensembles without detailing new axioms or parameters.

axioms (1)

domain assumption Principle of equivalence of ensembles applies to ZLRRs
Invoked to extend Gaussian and geometric properties to a wide class of zero-coefficient sequences

pith-pipeline@v0.9.0 · 5529 in / 1217 out tokens · 27245 ms · 2026-05-10T09:41:47.288324+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

J., Tosteson, P

Bower, A., Insoft, R., Li, S., Miller, S. J., Tosteson, P. (2015). The distribution of gaps between summands in generalized Zeckendorf decompositions.J. Combin. Theory Ser. A. 135:130–160. DOI: 10.1016/j.jcta.2015.06.006

work page doi:10.1016/j.jcta.2015.06.006 2015
[2]

J., Siktar, J

Chen, E., Chen, R., Guo, L., Jiang, C., Miller, S. J., Siktar, J. M., Yu, P. (2019). Gaussian Behavior in Zeckendorf Decompositions from Lattices.Fibonacci Quart. 57(3):203–215

work page 2019
[3]

Di¸ skaya, O., Menken, H. (2021). Some Properties of the Plastic Constant.J. Sci. Arts 21(4):883–894

work page 2021
[4]

J., Moon, D., Varma, U

Demontigny, P., Do, T., Kulkarni, A., Miller, S. J., Moon, D., Varma, U. (2014). Generalizing Zeckendorf’s theorem to f-decompositions.J. Number Theory. 141:136–

work page 2014
[5]

DOI: 10.1016/j.jnt.2014.02.012

work page doi:10.1016/j.jnt.2014.02.012 2014
[6]

E., Gray, L

Fristedt, B. E., Gray, L. F. (1996).A modern approach to probability theory. Boston: Birkh¨ auser. Theorems 14.15 and 18.21. ISBN 0-8176-3807-5

work page 1996
[7]

S., Miller, S

Kolo˘ glu, M., Kopp, G. S., Miller, S. J., Wang, Y. (2011). On the number of summands in Zeckendorf decompositions.Fibonacci Quart.49(2):116–130

work page 2011
[8]

Lekkerkerker, C. G. (1952). Voorstelling van natuurlijke getallen door een som van getallen van Fibonacci.Simon Stevin. 29:190–195

work page 1952
[9]

(1925).Calcul des probabilit´ es

L´ evy, P. (1925).Calcul des probabilit´ es. Gauthier-Villars, Paris

work page 1925
[10]

Li, R., Miller, S. J. (2019). Central Limit Theorems for gaps of generalized Zeckendorf decompositions.Fibonacci Quart.57(5):1–32. 20 S. SALAMI

work page 2019
[11]

C., Miller, S

Martinez, T. C., Miller, S. J., Mizgerd, C., Sun, C. (2022). Generalizing Zeckendorf’s Theorem to Homogeneous Linear Recurrences, I.arXiv preprint arXiv:2001.08455

work page arXiv 2022
[12]

J., Schiffman, D

Miller, S. J., Schiffman, D. (2016). The distribution of the number of summands in far-difference representations.Fibonacci Quart.54(5):129–149

work page 2016
[13]

J., Wang, Y

Miller, S. J., Wang, Y. (2012). From Fibonacci numbers to Central Limit Type The- orems.J. Combin. Theory Ser. A. 119(7):1398–1413. DOI: 10.1016/j.jcta.2012.02.001

work page doi:10.1016/j.jcta.2012.02.001 2012
[14]

Zeckendorf, E. (1972). Repr´ esentation des nombres naturels par une somme de nom- bres de Fibonacci ou de nombres de Lucas.Bull. Soc. Roy. Sci. Li` ege. 41:179–182. MSC2020: 11B39, 11K65 Institute of Mathematics and Statistics, Rio de Janeiro State University, Maracan˜a, Rio de Janeiro, 20950-000, RJ, Brazil Email address:Sajad.salami@ime.uerj.br

work page 1972

[1] [1]

J., Tosteson, P

Bower, A., Insoft, R., Li, S., Miller, S. J., Tosteson, P. (2015). The distribution of gaps between summands in generalized Zeckendorf decompositions.J. Combin. Theory Ser. A. 135:130–160. DOI: 10.1016/j.jcta.2015.06.006

work page doi:10.1016/j.jcta.2015.06.006 2015

[2] [2]

J., Siktar, J

Chen, E., Chen, R., Guo, L., Jiang, C., Miller, S. J., Siktar, J. M., Yu, P. (2019). Gaussian Behavior in Zeckendorf Decompositions from Lattices.Fibonacci Quart. 57(3):203–215

work page 2019

[3] [3]

Di¸ skaya, O., Menken, H. (2021). Some Properties of the Plastic Constant.J. Sci. Arts 21(4):883–894

work page 2021

[4] [4]

J., Moon, D., Varma, U

Demontigny, P., Do, T., Kulkarni, A., Miller, S. J., Moon, D., Varma, U. (2014). Generalizing Zeckendorf’s theorem to f-decompositions.J. Number Theory. 141:136–

work page 2014

[5] [5]

DOI: 10.1016/j.jnt.2014.02.012

work page doi:10.1016/j.jnt.2014.02.012 2014

[6] [6]

E., Gray, L

Fristedt, B. E., Gray, L. F. (1996).A modern approach to probability theory. Boston: Birkh¨ auser. Theorems 14.15 and 18.21. ISBN 0-8176-3807-5

work page 1996

[7] [7]

S., Miller, S

Kolo˘ glu, M., Kopp, G. S., Miller, S. J., Wang, Y. (2011). On the number of summands in Zeckendorf decompositions.Fibonacci Quart.49(2):116–130

work page 2011

[8] [8]

Lekkerkerker, C. G. (1952). Voorstelling van natuurlijke getallen door een som van getallen van Fibonacci.Simon Stevin. 29:190–195

work page 1952

[9] [9]

(1925).Calcul des probabilit´ es

L´ evy, P. (1925).Calcul des probabilit´ es. Gauthier-Villars, Paris

work page 1925

[10] [10]

Li, R., Miller, S. J. (2019). Central Limit Theorems for gaps of generalized Zeckendorf decompositions.Fibonacci Quart.57(5):1–32. 20 S. SALAMI

work page 2019

[11] [11]

C., Miller, S

Martinez, T. C., Miller, S. J., Mizgerd, C., Sun, C. (2022). Generalizing Zeckendorf’s Theorem to Homogeneous Linear Recurrences, I.arXiv preprint arXiv:2001.08455

work page arXiv 2022

[12] [12]

J., Schiffman, D

Miller, S. J., Schiffman, D. (2016). The distribution of the number of summands in far-difference representations.Fibonacci Quart.54(5):129–149

work page 2016

[13] [13]

J., Wang, Y

Miller, S. J., Wang, Y. (2012). From Fibonacci numbers to Central Limit Type The- orems.J. Combin. Theory Ser. A. 119(7):1398–1413. DOI: 10.1016/j.jcta.2012.02.001

work page doi:10.1016/j.jcta.2012.02.001 2012

[14] [14]

Zeckendorf, E. (1972). Repr´ esentation des nombres naturels par une somme de nom- bres de Fibonacci ou de nombres de Lucas.Bull. Soc. Roy. Sci. Li` ege. 41:179–182. MSC2020: 11B39, 11K65 Institute of Mathematics and Statistics, Rio de Janeiro State University, Maracan˜a, Rio de Janeiro, 20950-000, RJ, Brazil Email address:Sajad.salami@ime.uerj.br

work page 1972