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

This paper demonstrates that despite the loss of unique decomposition in Zero Linear Recurrence Relations (ZLRRs) like the Lagonacci sequence, the number of summands in canonical greedy decompositions still converges to a Gaussian distribution with geometrically decaying gaps, while the number of legal decompositions grows exponentially at a rate of $\alpha=2$ .

Original authors: Sajad Salami

Published 2026-04-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sajad Salami

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant pile of LEGO bricks. In the world of math, there's a famous rule called Zeckendorf's Theorem. It says that if you have a specific set of special LEGO bricks (based on the Fibonacci sequence), you can build any number you want, and you can do it in exactly one way if you follow a simple rule: you can't use two bricks that are right next to each other in the sequence.

Think of it like a unique password. No matter what number you want to represent, there is only one "legal" combination of bricks that works.

The New Twist: The "Broken" Rule

This paper asks a "What if?" question: What happens if we break the rule that guarantees uniqueness?

The author, S. Salami, looks at a slightly different set of bricks called the Lagonacci sequence. In this system, the "recipe" for making the next brick size is a bit weird: the next size depends on the one two steps back and the one three steps back, but it ignores the one immediately before it.

Because of this "missing link" in the recipe, the magic of uniqueness disappears. Now, a single number can be built in many different ways using these bricks. It's like having a lock that can be opened with dozens of different keys instead of just one.

The Big Surprise: Chaos vs. Order

Usually, when you break a rule like "uniqueness," you expect the whole system to fall into chaos. You'd expect the numbers to behave erratically, with no patterns.

But the paper discovers something amazing: Even though there are many ways to build the numbers, the statistical behavior remains surprisingly orderly.

Here are the two main discoveries, explained with analogies:

1. The "Crowd" Still Acts Like a Bell Curve

Imagine you ask 1,000 people to build the number 1,000,000 using these bricks.

In the old system (Unique): Everyone uses the exact same number of bricks.
In this new system (Non-unique): Everyone uses a different number of bricks. Some use 10, some use 12, some use 15.

You might think the number of bricks used would be all over the place. But the paper proves that if you graph the results, they form a perfect Bell Curve (a Gaussian distribution). Most people use a number of bricks right near the average, with fewer people using very few or very many.

The Analogy: It's like a crowded concert. Even though everyone is moving around freely (non-unique paths), the crowd still naturally clusters in the middle of the room. The "chaos" of having many choices actually smooths out into a predictable pattern.

2. The "Gaps" Shrink Like a Fading Echo

When you build a number, you pick specific bricks. The "gap" is the distance between the sizes of the bricks you picked.

The Discovery: The paper shows that small gaps are very common, but large gaps become rare very quickly.
The Analogy: Imagine you are skipping stones across a pond. It's easy to skip a stone a short distance. It's possible to skip it a long distance, but it gets exponentially harder the further you try to go. The paper proves that in this new system, the "skips" between your chosen bricks follow this exact same "fading echo" pattern.

The "Explosion" of Options

There is one major difference between the old system and this new one: Quantity.

Old System: 1 Number = 1 Way to build it.
New System: 1 Number = Thousands of ways to build it.

The paper calculates that as the numbers get bigger, the number of possible ways to build them explodes exponentially. It grows much faster than the numbers themselves.

The Analogy: Think of a maze.

In the old system, there is only one path to the exit.
In this new system, there are millions of paths.
The Twist: Even though there are millions of paths, if you pick a random path, the average length of the path and the pattern of turns still follow the same predictable rules as the single-path maze.

Why Does This Matter?

This is a big deal for mathematicians because it changes how we think about "order."

For a long time, mathematicians thought that Uniqueness was the secret sauce that created these beautiful statistical patterns (like the Bell Curve). This paper proves that Uniqueness is not necessary.

The patterns are actually caused by the underlying structure of the numbers themselves (the "recipe" for the bricks), not by the rule that says "there is only one way." Even when you allow chaos (multiple ways), the deep mathematical structure forces the system to behave in an orderly, predictable way.

Summary

The Problem: We broke the rule that makes number representations unique.
The Fear: We thought the math would become messy and unpredictable.
The Result: The math is still beautifully predictable! The number of bricks used follows a Bell Curve, and the gaps between them shrink predictably.
The Catch: There are now many ways to build the same number, and the number of ways grows explosively fast.

In short: Even when you give a system too many choices, the universe still finds a way to keep things statistically neat.

1. Problem Statement

The paper addresses a significant gap in additive number theory regarding Zero Linear Recurrence Relations (ZLRRs).

Context: Zeckendorf's theorem and its generalization to Positive Linear Recurrence Sequences (PLRS) establish that every positive integer has a unique representation as a sum of non-consecutive terms. In PLRS, the recurrence coefficients are strictly positive ( $c_i > 0$ ).
The Gap: Existing literature relies heavily on the "positivity" of coefficients to ensure uniqueness. This paper investigates what happens when the leading coefficient is zero ( $c_1 = 0$ ), creating a ZLRR.
Core Question: In systems where uniqueness is lost (i.e., an integer can be represented in multiple ways), do the fundamental statistical regularities observed in PLRS—specifically the Gaussian distribution of the number of summands and the geometric decay of gap distributions—persist?
Case Study: The author focuses on the Lagonacci sequence ( $Z_{n+1} = Z_{n-1} + Z_{n-2}$ ), which corresponds to the plastic ratio ( $\lambda_1 \approx 1.3247$ ).

2. Methodology

The paper employs a combination of combinatorial analysis, generating functions, probability theory, and statistical mechanics principles.

Greedy Decomposition: Since uniqueness is lost, the study focuses on the canonical greedy decomposition (repeatedly subtracting the largest possible term $Z_k \le m$ ). While legal decompositions are non-unique, the greedy algorithm produces a unique path for any integer.
Generating Functions & Recurrence: The author constructs generating functions ( $G_N(y)$ ) to count the number of summands. By leveraging the recurrence relation, they derive recursive formulas for these functions to compute moments (mean and variance).
Characteristic Functions & Lévy's Theorem: To prove Gaussian convergence, the paper utilizes Lévy's Continuity Theorem. It analyzes the characteristic functions of the normalized number of summands, showing that higher-order cumulants vanish as $N \to \infty$ , leaving the standard normal characteristic function ( $e^{-t^2/2}$ ).
Transfer Matrix Method: To quantify the non-uniqueness, the author models the set of legal decompositions as binary strings avoiding a specific forbidden substring (derived from the recurrence). A transfer matrix is constructed to count these strings, and its Perron-Frobenius eigenvalue determines the growth rate.
Equivalence of Ensembles: A key theoretical tool is the principle of equivalence of ensembles (borrowed from statistical mechanics). The author proves that the microcanonical ensemble (uniform distribution over integers in $[Z_N, Z_{N+1})$ ) and the canonical ensemble (stationary measure on infinite digit sequences weighted by the dominant root) yield identical limit laws for local statistics. This allows the transfer of results from the easier-to-analyze canonical setting to the uniform integer setting.

3. Key Contributions and Results

A. Persistence of Gaussian Behavior

Despite the loss of uniqueness, the number of summands in the greedy decomposition of the Lagonacci sequence converges to a Gaussian distribution.

Theorem 3.6: For integers chosen uniformly from $[Z_N, Z_{N+1})$ $[Z_{N}, Z_{N + 1})$ , the number of summands $K_N$ $K_{N}$ satisfies:
- Mean: $E[K_N] = C_1 N + O(1)$
- Variance: $Var(K_N) = C_2 N + O(1)$
- Normalized variable $X_N \xrightarrow{d} \mathcal{N}(0, 1)$ .
Significance: This proves that Gaussianity is a fundamental property of the linear recurrence structure and the dominant root, not merely a consequence of unique representations.

B. Geometric Decay of Gaps

The distribution of gaps between the indices of consecutive summands in the greedy decomposition decays geometrically.

Theorem 3.7: The limiting probability of a gap of length $k$ ( $k \ge 2$ ) is $P(g=k) = C \lambda_1^{-k}$ .
Specifics: For the Lagonacci sequence, gaps of length 0 and 1 have specific non-zero probabilities, but for $k \ge 2$ , the decay is governed by the plastic ratio $\lambda_1$ .

C. Quantification of Non-Uniqueness

The paper rigorously quantifies the explosion of legal representations.

Theorem 3.9: The total number of legal decompositions for integers up to $Z_N$ grows as $D(Z_N) \sim C \cdot 2^N$ .
Growth Rate Comparison: The underlying sequence grows as $\lambda_1^N$ ( $\lambda_1 \approx 1.32$ ), while the number of representations grows as $2^N$ .
Average Redundancy: The average number of decompositions for a typical integer of size $Z_N$ grows exponentially at a rate of $(2/\lambda_1)^N \approx (1.509)^N$ . This indicates a massive redundancy compared to the unique representation in PLRS.

D. Generalization to ZLRRs

Theorem 3.8 & 4.9: The results are not limited to the Lagonacci sequence. They extend to any ZLRR with a characteristic polynomial having a unique, simple real dominant root. The Equivalence of Ensembles (Theorem 4.7) ensures that local statistics (like gap distributions and summand counts) behave identically under uniform sampling and the canonical stationary measure.

4. Significance and Implications

Decoupling Uniqueness from Statistics: The paper fundamentally challenges the assumption that unique representations are necessary for statistical regularity. It demonstrates that Gaussianity and geometric gaps are robust features of linear recurrences, persisting even when the combinatorial structure becomes non-unique.
Theoretical Bridge: By applying the Equivalence of Ensembles to number theory, the paper provides a rigorous framework for analyzing non-unique numeration systems, bridging the gap between combinatorial number theory and statistical mechanics.
New Class of Systems: It defines and characterizes ZLRRs as a distinct class of numeration systems where the "cost" of losing uniqueness is an exponential explosion in the number of representations, yet the "average" behavior remains predictable and Gaussian.
Future Directions: The work opens avenues for studying the variance of the number of decompositions (beyond the logarithmic average) and extending these methods to recurrences with multiple dominant roots or higher-dimensional lattices.

In summary, Salami's work proves that the statistical laws governing Zeckendorf-type decompositions are deeper than the uniqueness property itself, relying instead on the algebraic dominance of the recurrence's characteristic root.