  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.
