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Zeckendorf's Theorem states that any positive integer can be written uniquelyas a sum of non-adjacent Fibonacci numbers. We consider higher-dimensionallattice analogues, where a legal decomposition of a number $n$ is a collectionof lattice points such that each point is included at most once. Once a pointis chosen, all future points must have strictly smaller coordinates, and thepairwise sum of the values of the points chosen equals $n$. We prove that thedistribution of the number of summands in these lattice decompositionsconverges to a Gaussian distribution in any number of dimensions. As animmediate corollary we obtain a new proof for the asymptotic number of certainlattice paths.

Gaussian Behavior in Zeckendorf Decompositions From Lattices