[Paper Review] Gaussian Behavior in Zeckendorf Decompositions From Lattices

Eric Chen, Robin Chen+5 more|arXiv (Cornell University)|Sep 16, 2018

Mathematical Dynamics and Fractals13 references4 citations

TL;DR

This paper establishes that the number of summands in generalized Zeckendorf decompositions over d-dimensional lattices converges to a Gaussian distribution as the lattice size increases. By modeling legal decompositions as simple jump paths on ordered lattice points, the authors prove central limit behavior for the count of summands, extending classical Zeckendorf results to higher dimensions with rigorous asymptotic analysis of mean and variance.

ABSTRACT

Zeckendorf's Theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. We consider higher-dimensional lattice analogues, where a legal decomposition of a number $n$ is a collection of lattice points such that each point is included at most once. Once a point is chosen, all future points must have strictly smaller coordinates, and the pairwise sum of the values of the points chosen equals $n$. We prove that the distribution of the number of summands in these lattice decompositions converges to a Gaussian distribution in any number of dimensions. As an immediate corollary we obtain a new proof for the asymptotic number of certain lattice paths.

Motivation & Objective

• To extend Zeckendorf's theorem on unique non-adjacent Fibonacci sum decompositions to higher-dimensional lattices.
• To define legal decompositions in d-dimensional lattices where each point is used at most once and coordinates strictly decrease along the path.
• To analyze the distribution of the number of summands in such lattice decompositions and prove convergence to a Gaussian distribution.
• To generalize one-dimensional Zeckendorf decomposition results to truly multidimensional settings using lattice path combinatorics.
• To provide a new proof for the asymptotic number of certain lattice paths using the derived Gaussian behavior.

Proposed method

• Define d-dimensional lattice decompositions as finite sets of lattice points where each point is used at most once and all subsequent points have strictly smaller coordinates in every dimension.
• Introduce 'simple jump paths' as sequences of lattice points with strictly decreasing coordinates, forming the basis for legal decompositions.
• Assign values to lattice points via a recursive construction analogous to the one-dimensional simple Zeckendorf sequence, ensuring uniqueness of decomposition.
• Use combinatorial enumeration of paths to compute the mean and variance of the number of summands in decompositions.
• Apply moment-generating function techniques and asymptotic analysis of binomial sums to establish convergence to a normal distribution.
• Derive exact formulas for mean and standard deviation in 1D and 2D cases using identities involving central binomial coefficients and symmetric sums.

Experimental results

Research questions

• RQ1 Does the number of summands in d-dimensional lattice decompositions converge to a Gaussian distribution as the lattice size grows?
• RQ2 How do the mean and variance of the number of summands scale with the size of the lattice in d dimensions?
• RQ3 Can the asymptotic number of certain lattice paths be derived from the Gaussian behavior of decomposition summand counts?
• RQ4 What is the precise asymptotic behavior of the mean and standard deviation of summand counts in two-dimensional simple jump paths?
• RQ5 How does the combinatorial structure of lattice paths affect the distribution of summands in higher-dimensional Zeckendorf decompositions?

Key findings

• The distribution of the number of summands in d-dimensional lattice decompositions converges to a Gaussian distribution as the lattice size increases.
• In two dimensions, the mean number of summands in a simple jump path from (n,n) to the origin is asymptotically (1/2)n + 1.
• The standard deviation of the number of summands in two-dimensional simple jump paths is asymptotically n/(2√(2n−1)).
• The authors derive exact expressions for the mean and variance of summand counts in 1D and 2D cases using symmetric sums of binomial coefficients.
• The proof technique relies on combinatorial identities involving ∑k²(n choose k)² = n²(2n−2 choose n−1), which holds only in two dimensions.
• The result provides a new proof for the asymptotic number of certain lattice paths by linking path enumeration to the Gaussian limit of summand counts.