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Gaps of Summands of the Zeckendorf Lattice

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Pith's one-line read In the two-dimensional Zeckendorf lattice, a randomly chosen gap vector $(v_1,v_2)$ between consecutive summands has limiting probability $1/2^{v_1+v_2}$, a bivariate geometric law.

desk verdict Clean, correct limit theorem for 2D lattice gap vectors; modest but genuine extension of the Zeckendorf program. read the letter →

arxiv 1909.01935 v1 pith:BWVUCX2A submitted 2019-09-04 math.NT math.CO

Neelima Borade , Dexter Cai , David Z. Chang , Bruce Fang , Alex Liang , Steven J. Miller , Wanqiao Xu This is my paper · ORCID
classification math.NTmath.CO MSC 11B0205A02
keywords Zeckendorfdecompositionssimplejumppathtwo-dimensionallatticegapvectorsbivariategeometricdistributionVandermonde'sidentitylegaldecompositionbinomialcoefficients
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The pith

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

The reading

Zeckendorf's theorem says every positive integer has a unique decomposition into non-adjacent Fibonacci numbers, and for generalized one-dimensional recurrences the gaps between summands decay geometrically. This paper asks whether the same gap behavior survives when decompositions are two-dimensional lattices, where a legal decomposition is a chain of lattice points moving strictly down and left. The central result is that for fixed positive integers $v_1$ and $v_2$, among all such simple jump paths from $(n+1,n+1)$ to the origin, the probability that a randomly chosen gap vector equals $(v_1,v_2)$ converges to $1/2^{v_1+v_2}$ as $n$ tends to infinity. That is a bivariate geometric limit, the exact two-dimensional analogue of the one-dimensional geometric decay, and it yields a limiting gap-sum distribution of $(v-1)/2^v$. The proof is combinatorial, using Vandermonde's identity to count paths and Stirling's formula to extract the limit, so the result extends the known one-dimensional theory to a genuinely multidimensional setting.

What carries the argument

The central object is the simple jump path: a sequence of lattice points from a starting corner to the origin in which every coordinate strictly decreases at each step, so each step is a legal decomposition move in the $d$-dimensional Zeckendorf lattice. In two dimensions the number of such paths from $(a_1,a_2)$ to the origin is the binomial coefficient $\binom{a_1+a_2-2}{a_1-1}$, obtained from Vandermonde's identity. The gap count splits each path at the two endpoints of a proposed gap: the number of paths containing a gap vector $(v_1,v_2)$ is a product of the number of paths from the lower endpoint to the origin and the number from the upper endpoint to the far corner, summed over all interior positions, with separate corner terms at $(0,0)$ and $(n-v_1+1,n-v_2+1)$. Stirling's formula evaluates the resulting ratio.

What would settle it

Enumerate all simple jump paths from $(11,11)$ to $(0,0)$ and compare the number of gap vectors equal to $(1,1)$ divided by the total number of gap vectors with the value $316,030/1,108,536$ predicted by Lemmas 3.2 and 3.3; if the counts disagree, the boundary-case counting is wrong, since the theorem says the ratio converges to $1/4$ (and generally to $1/2^{v_1+v_2}$).

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Extended reading notes

Core claim

The authors prove Theorem 1.3: start at $(n+1,n+1)$ and consider every simple jump path to $(0,0)$, where each step strictly decreases both coordinates; adding the origin as a final point just adds a negligible extra gap. For fixed positive integers $v_1,v_2$, the probability that a uniformly chosen gap vector among all such paths equals $(v_1,v_2)$ satisfies $P(v_1,v_2)=g(n+1;(v_1,v_2))/g_2(n+1)\to 1/2^{v_1+v_2}$. The numerator is counted in Lemma 3.2 as $g(n+1;(v_1,v_2)) = (2n-v_1-v_2-1)\binom{2n-v_1-v_2-2}{n-v_1-1} + 2\binom{2n-v_1-v_2}{n-v_1}$, built from interior placements of the gap plus two corner placements at $(0,0)$ and the far corner, and the denominator is $g_2(n+1)=(n/2+1)\binom{2n}{n}$, the total number of gap vectors. Stirling's approximation then drives the ratio to $1/2^{v_1+v_2}$. The immediate corollary (Theorem 1.4) is that the limiting probability that a gap sum equals $v\ge 2$ is $(v-1)/2^v$, since exactly $v-1$ positive pairs sum to $v$.

Load-bearing premise

The counting assumes that a simple jump path never visits a lattice point with one coordinate zero except the final origin, so the only boundary placements of a gap are the two corners; if paths could touch an axis, additional boundary terms would enter the gap count and could change the limiting distribution.

Editorial extensions

If this is right

  • For fixed $v_1,v_2$, the limiting gap-vector probabilities are $2^{-(v_1+v_2)}$, so the two coordinates converge to independent geometric variables with parameter $1/2$.
  • The limiting gap-sum distribution is $P(v)=(v-1)/2^v$ for $v\ge 2$, with $P(0)=P(1)=0$.
  • In the limit the expected gap sum is $4$, since each coordinate of the bivariate geometric law has mean $2$; small gaps dominate, with gap vector $(1,1)$ occurring with probability $1/4$.
  • Because the proof of the path count relies on Vandermonde's identity, the same argument does not directly extend to dimension $d\ge 3$; the authors note that new combinatorial identities would be needed.
  • The result matches the one-dimensional geometric decay established for generalized Zeckendorf decompositions, showing that the two-dimensional lattice preserves this phenomenon even though unique decomposition is lost.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the same limit were sought for compound paths (steps allowed to move down, left, or diagonally), paths may touch an axis away from the origin; the additional boundary terms are exactly where the gap law could deviate from $1/2^{v_1+v_2}$.
  • For Euclidean gap distance, the limiting squared-distance probability would be $\sum_{v_1^2+v_2^2=g} 2^{-(v_1+v_2)}$, a sum over representations of $g$ as a sum of two squares; the paper only discusses this qualitatively.
  • A relaxed model with weak decrease ($i'_j \le i_j$) would allow axis-touching paths and could change the limiting distribution, so the strict-decrease convention is not a mere technical convenience but part of what the theorem asserts.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. The paper studies the distribution of gap vectors between consecutive points in legal decompositions (simple jump paths) on the two-dimensional Zeckendorf lattice. Working with paths from (n+1,n+1) to (0,0), the authors count the total number of occurrences of a fixed gap vector (v1,v2) by splitting on the location of the gap's lower endpoint, obtaining a closed-form binomial expression. Dividing by the total number of gap vectors, which is computed in two independent ways, and applying Stirling's formula, they prove Theorem 1.3: the probability that a uniformly chosen gap vector equals (v1,v2) tends to 1/2^{v1+v2}, i.e., a product of two independent geometric(1/2) distributions. Theorem 1.4 derives the limiting gap-sum distribution (v-1)/2^v. The paper also discusses the difficulty of extending the argument to d>=3 and outlines conjectures about gap distances and longest gaps.

Significance. If correct, this is a natural and nontrivial extension of the one-dimensional gap-decay results to a higher-dimensional lattice setting, and the clean bivariate geometric limit is a nice addition to the Zeckendorf literature. The proof is elementary but careful: the boundary case analysis in Lemma 3.2 is the delicate point, and it is justified by the strict-decrease property that rules out axis points except the origin; the concern that additional boundary terms might enter therefore does not land on reading the paper. I also checked the Stirling computation in the proof of Theorem 1.3 and the two proofs of Lemma 3.3 for internal consistency, and the combinatorial identities are sound. The paper is explicit about the limitations of the method for d>=3 and provides a self-contained second proof of the key total-count lemma, which strengthens the exposition.

minor comments (6)
  1. [Section 3, near (3.1)] The remark that the extra origin gap 'can thus be safely ignored' is confusing because the proof actually counts it as Case (2) in Lemma 3.2; please clarify that the origin gap is included in the counting and that its contribution vanishes in the limit, or alternatively state that one may subtract it from both numerator and denominator without changing the limit.
  2. [Proof of Theorem 1.3, (3.22)] The total count is denoted g(n+1;(v1,v2)) in Lemma 3.2 but is written as G(n+1;(v1,v2)) in the proof, which collides with the per-placement notation G((x,y),(x+v1,y+v2)) introduced in Lemma 3.1; please use consistent notation.
  3. [Theorems 1.3 and 1.4] Please state explicitly the probability model: a gap vector is chosen uniformly among all gap vectors occurring in all simple jump paths, or equivalently a path is chosen uniformly and then one of its gap vectors is chosen uniformly; without this, 'the probability that a gap vector equals (v1,v2)' is ambiguous.
  4. [Section 2, Definition of simple jump paths] The sentence 'for any choice of starting point (a1,...,ad) in R^d' should say N^d or N_+^d, since the coordinates are integer lattice points.
  5. [Lemma 3.2] For fixed positive integers (v1,v2), the binomial expressions implicitly assume n is large enough (roughly n>v1+v2); stating this explicitly would avoid evaluating binomial coefficients with negative entries.
  6. [Throughout] There are several typographical errors, for example 'shoud l' in the first proof of Lemma 3.3 and 'LA TTICE' in the paper header; a careful proofreading pass is advised.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main gap-vector limit is derived from first-principles combinatorial counting.

full rationale

The paper's central claim (Theorem 1.3) is obtained by direct combinatorial enumeration: Lemma 3.1 decomposes a gap-vector count into a product of two simple-jump-path counts, Lemma 3.2 sums these counts over the admissible interior and boundary positions (with the boundary restriction following from the strict-decrease definition reproduced in Section 2 from [CCGJMSY]), and Lemma 3.3 supplies the total number of gap vectors. The only self-citation appears in the First Proof of Lemma 3.3, which borrows the mean step count mu_2(n+1)=n/2+1 from [CCGJMSY]; this dependence is not load-bearing because the immediately following Second Proof derives the same total self-containedly from standard binomial identities, and Vandermonde's identity is proved in Lemma 2.1. No fitted parameter is introduced, no target distribution is assumed, and Stirling's approximation is applied only after exact identities and only for fixed (v1,v2). The boundary-case assumption in Lemma 3.2 follows from the path model's strict-decrease condition rather than being an input tailored to the result, and the paper explicitly flags higher-dimensional and longest-gap questions as open rather than importing them from prior work. The derivation is therefore self-contained, with no circular reduction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or invented entities. The derivation relies on standard combinatorial identities and on the path model adopted from prior work. The mean-step count from [CCGJMSY] appears only in an alternate proof of Lemma 3.3; the second proof is self-contained.

assumptions (4)
  • standard math Vandermonde's identity
    Used in Lemma 2.1 and in the proof of Lemma 3.2 to collapse the double sum over x and y into a closed form.
  • standard math Stirling's approximation
    Used in the proof of Theorem 1.3 to evaluate the limit of the ratio of factorials.
  • domain assumption Definition of legal decomposition as strict-decrease simple jump paths from [CCGJMSY]
    The entire paper operates within this path model; if the model were relaxed, the gap counts and the limiting distribution could change.
  • domain assumption Mean number of steps formula from [CCGJMSY, Lemma 3.1]
    Used only in the first proof of Lemma 3.3; the second proof of that lemma is self-contained, so this axiom is not load-bearing.

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Pith. "Pith review of Gaps of Summands of the Zeckendorf Lattice." pith.science (2026). https://pith.science/paper/BWVUCX2A

@misc{pith2026190901935,
  author       = {Pith},
  title        = {Pith review of: Gaps of Summands of the Zeckendorf Lattice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BWVUCX2A}},
  note         = {Machine review of arXiv:1909.01935}
}
read the original abstract

A beautiful theorem of Zeckendorf states that every positive integer has a unique decomposition as a sum of non-adjacent Fibonacci numbers. Such decompositions exist more generally, and much is known about them. First, for any positive linear recurrence {Gn} the number of summands in the legal decompositions for integers in [Gn, Gn+1) converges to a Gaussian distribution. Second, Bower, Insoft, Li, Miller, and Tosteson proved that the probability of a gap between summands in a decomposition which is larger than the recurrence length converges to geometric decay. While most of the literature involves one-dimensional sequences, some recent work by Chen, Guo, Jiang, Miller, Siktar, and Yu have extended these decompositions to d-dimensional lattices, where a legal decomposition is a chain of points such that one moves in all d dimensions to get from one point to the next. They proved that some but not all properties from 1-dimensional sequences still hold. We continue this work and look at the distribution of gaps between terms of legal decompositions, and prove similar to the 1-dimensional cases that when d = 2 the gap vectors converge to a bivariate geometric random variable.

Figures

Figures reproduced from arXiv: 1909.01935 by the authors.

Figure 1
Figure 1. Set-up to compute the number of simple jump paths from (0, 0) to (n + 1, n + 1) with a gap of (v1, v2) starting at (x, y). Counting the number of such paths is the same as counting the number of paths in the bottom left rectangle and multiplying by the number of paths in the top right. Proof. For ease of calculation, we break into three cases: (1) 1 ≤ x ≤ n − v1 and 1 ≤ y ≤ n − v2, (2) x = 0 and y = 0, (3) x = n − v… view at source ↗

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Works this paper leans on

36 extracted references · 35 canonical work pages

  1. [1]

    Beckwith, A

    O. Beckwith, A. Bower, L. Gaudet, R. Insoft, S. Li, S. J. Miller and P. Tosteson, The Average Gap Distribution for Generalized Zeckendorf Decompositions, Fibonacci Quarterly 51 (2013), 13--27

    work page 2013

  2. [2]

    Ben-Ari, S

    I. Ben-Ari, S. Miller, A Probabilistic Approach to Generalized Zeckendorf Decompositions, SIAM Journal on Discrete Mathematics, 30 (2016), no. 2, 1302--1332

    work page 2016

  3. [3]

    A. Best, P. Dynes, X. Edelsbrunner, B. McDonald, S. Miller, K. Tor, C. Turnage-Butterbaugh, M. Weinstein, Gaussian Behavior of the Number of Summands in Zeckendorf Decompositions in Small Intervals, Fibonacci Quarterly, 52 (2014), no. 5, 47--53

    work page 2014

  4. [4]

    Bower, R

    A. Bower, R. Insoft, S. Li, S. Miller, P. Tosteson, The Distribution of Gaps Between Summands in Generalized Zeckendorf Decompositions, Journal of Combinatorial Theory, 135 (2015), 130--160

    work page 2015

  5. [5]

    J. L. Brown, Jr., Zeckendorf's Theorem and Some Applications, The Fibonacci Quarterly, Vol. 2, No. 3 (Oct. 1964), pages 163--168

    work page 1964

  6. [6]

    E. Chen, R. Chen, L. Guo, C. Jiang, S. J. Miller, J. M. Siktar and Peter Yu, Gaussian Behavior in Zeckendorf Decompositions From Lattices, to appear in the Fibonacci Quarterly

    work page

  7. [7]

    Catral, P

    M. Catral, P. Ford, P. Harris, S. Miller, D. Nelson, Generalizing Zeckendorf's Theorem: The Kentucky Sequence, Fibonacci Quarterly, 52 (2014), no. 5, 68--90

    work page 2014

  8. [8]

    Catral, P

    M. Catral, P. Ford, P. E. Harris, S. J. Miller, and D. Nelson, Legal Decompositions Arising from Non-positive Linear Recurrences, Fibonacci Quarterly 54 (2016), no. 4, 3448--365

    work page 2016

Show all 36 references
  1. [9]

    Catral, P

    M. Catral, P. Ford, P. E. Harris, S. J. Miller, D. Nelson, Z. Pan and H. Xu, New Behavior in Legal Decompositions Arising from Non-positive Linear Recurrences, Fibonacci Quarterly 55 (2017), no. 3, 252--275 (expanded arXiv version: http://arxiv.org/pdf/1606.09309 )

    2017 arXiv

  2. [10]

    D. E. Daykin, Representation of Natural Numbers as Sums of Generalized Fibonacci Numbers, J. London Mathematical Society 35 (1960), 143--160

    1960

  3. [11]

    Demontigny, T

    P. Demontigny, T. Do, A. Kulkarni, S. Miller, D. Moon, U. Varma, Generalizing Zeckendorf's Theorem to f-Decompositions, Journal of Number Theory, 141 (2014), 136--158

    2014

  4. [12]

    Dorward, P

    R. Dorward, P. Ford, E. Fourakis, P. Harris, S. Miller, E. Palsson, H. Paugh, New Behavior in Legal Decompositions Arising From Non-Positive Linear Recurrences, Fibonacci Quarterly, 55 (2017), no. 3, 252--275

    2017

  5. [13]

    Drmota and J

    M. Drmota and J. Gajdosik, The distribution of the sum-of-digits function, J. Th\'eor. Nombr\'es Bordeaux 10 (1998), no. 1, 17--32

    1998

  6. [14]

    Eger, Stirling's Approximation for Central Extended Binomial Coefficients, American Mathematical Monthly, 121 (2014), no

    S. Eger, Stirling's Approximation for Central Extended Binomial Coefficients, American Mathematical Monthly, 121 (2014), no. 4, 344--349

    2014

  7. [15]

    E. Fang, J. Jenkins, Z. Lee, D. Li, E. Lu, S. J. Miller, D. Salgado and Joshua Siktar Central Limit Theorems for Compound Paths on the 2-Dimensional Lattice, preprint 2019. https://arxiv.org/abs/1906.10645

    2019 arXiv

  8. [16]

    Filipponi, P

    P. Filipponi, P. J. Grabner, I. Nemes, A. Peth\"o, and R. F. Tichy, Corrigendum to: ``Generalized Zeckendorf expansions'', Appl. Math. Lett., 7 (1994), no. 6, 25--26

    1994

  9. [17]

    A. S. Fraenkel, Systems of Numeration, Amer. Math. Monthly 92 (1985), no. 2, 105--114

    1985

  10. [18]

    Gordon, M

    L. Gordon, M. F. Schilling, and M. S. Waterman, An extreme value theory for long head runs, Probability Theory and Related Fields 72 (1986) 279--287

    1986

  11. [19]

    P. J. Grabner, R. F. Tichy, I. Nemes, and A. Peth\"o, Generalized Zeckendorf expansions, Appl. Math. Lett. 7 (1994), no. 2, 25--28

    1994

  12. [20]

    Hamlin, Representing Positive Integers as a Sum of Linear Recurrence Sequences, Fibonacci Quarterly 50 (2012), no

    N. Hamlin, Representing Positive Integers as a Sum of Linear Recurrence Sequences, Fibonacci Quarterly 50 (2012), no. 2, 99--105

    2012

  13. [21]

    V. E. Hoggatt, Generalized Zeckendorf theorem, Fibonacci Quarterly 10 (1972), no. 1 (special issue on representations), pages 89--93

    1972

  14. [22]

    Hamlin and W

    N. Hamlin and W. A. Webb, Representing positive integers as a sum of linear recurrence sequences, Fibonacci Quarterly 50 (2012), no. 2, 99--105

    2012

  15. [23]

    T. J. Keller, Generalizations of Zeckendorf's theorem, Fibonacci Quarterly 10 (1972), no. 1 (special issue on representations), pages 95--102

    1972

  16. [24]

    Kologlu, G

    M. Kologlu, G. Kopp, S. Miller, Y. Wang, On the Number of Summands in Zeckendorf Decompositons, Journal of Number Theory, 49 (2011), no. 2, 116--130

    2011

  17. [25]

    Lamberger and J

    M. Lamberger and J. M. Thuswaldner, Distribution properties of digital expansions arising from linear recurrences, Math. Slovaca 53 (2003), no. 1, 1--20

    2003

  18. [26]

    C. G. Lekkerkerker, Voorstelling van natuurlyke getallen door een som van getallen van Fibonacci|, Simon Stevin 29 (1951-1952), 190--195

    1951

  19. [27]

    Lengyel, A Counting Based Proof of the Generalized Zeckendorf's Theorem, Fibonacci Quarterly 44 (2006), no

    T. Lengyel, A Counting Based Proof of the Generalized Zeckendorf's Theorem, Fibonacci Quarterly 44 (2006), no. 4, 324--325

    2006

  20. [28]

    Li and S

    R. Li and S. J. Miller, A Collection of Central Limit Type results in Generalized Zeckendorf Decompositions, the 17th International Fibonacci Conference, Fibonacci Quarterly 55 (2017), no. 5, 105--114

    2017

  21. [29]

    Li and S

    R. Li and S. J. Miller, Central Limit Theorems for Gaps of Generalized Zeckendorf Decompositions, to appear in the Fibonacci Quarterly. http://arxiv.org/abs/1606.08110v1.pdf

    arXiv

  22. [30]

    S. J. Miller, The Probability Lifesaver, Princeton University Press, 2017, 752 pages

    2017

  23. [31]

    Miller, Y

    S. Miller, Y. Wang, From Fibonacci Numbers to Central Limit Type Theorems, Journal of Combinatorial Theory, Series A 119 (2012), no. 7, 1398--1413

    2012

  24. [32]

    Miller, Y

    S. Miller, Y. Wang, Gaussian Behavior in Generalized Zeckendorf Decompositions, Combinatorial and Additive Number Theory, CANT 2011 and 2012 (Melvyn B. Nathanson, editor), Springer Proceedings in Mathematics & Statistics (2014), 159--173

    2014

  25. [33]

    M. F. Schilling, The Longest Run of Heads, The College Mathematics Journal 21 (1990), no. 3, 196--207

    1990

  26. [34]

    Steiner, Parry expansions of polynomial sequences, Integers, 2 (2002), Paper A14

    W. Steiner, Parry expansions of polynomial sequences, Integers, 2 (2002), Paper A14

    2002

  27. [35]

    Steiner, The Joint Distribution of Greedy and Lazy Fibonacci Expansions, Fibonacci Quarterly, 43 (2005), 60--69

    W. Steiner, The Joint Distribution of Greedy and Lazy Fibonacci Expansions, Fibonacci Quarterly, 43 (2005), 60--69

    2005

  28. [36]

    E. Zeckendorf, Repr\'esentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bulletin de la Soci\'et\'e Royale des Sciences de Li\'ege 41 (1972), pages 179--182

    1972

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