  A beautiful theorem of Zeckendorf states that every positive integer has a
unique decomposition as a sum of non-adjacent Fibonacci numbers. Such
decompositions exist more generally, and much is known about them. First, for
any positive linear recurrence {Gn} the number of summands in the legal
decompositions for integers in [Gn, Gn+1) converges to a Gaussian distribution.
Second, Bower, Insoft, Li, Miller, and Tosteson proved that the probability of
a gap between summands in a decomposition which is larger than the recurrence
length converges to geometric decay. While most of the literature involves
one-dimensional sequences, some recent work by Chen, Guo, Jiang, Miller,
Siktar, and Yu have extended these decompositions to d-dimensional lattices,
where a legal decomposition is a chain of points such that one moves in all d
dimensions to get from one point to the next. They proved that some but not all
properties from 1-dimensional sequences still hold. We continue this work and
look at the distribution of gaps between terms of legal decompositions, and
prove similar to the 1-dimensional cases that when d = 2 the gap vectors
converge to a bivariate geometric random variable.
