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arXiv · 1809.05829

Gaussian Behavior in Zeckendorf Decompositions From Lattices

Eric Chen·Robin Chen·Lucy Guo·Cindy Jiang·Steven J. Miller·Joshua M. Siktar·Peter Yu
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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.

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Eric Chen, Robin Chen, Lucy Guo, Cindy Jiang, Steven J. Miller, Joshua M. Siktar, Peter Yu. 2018-09-16. Gaussian Behavior in Zeckendorf Decompositions From Lattices. https://arxiv.org/abs/1809.05829

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