Individual Gap Measures from Generalized Zeckendorf Decompositions
Abstract
Zeckendorf's theorem states that every positive integer can be decomposed uniquely as a
sum of nonconsecutive Fibonacci numbers. The distribution of the number of summands
converges to a Gaussian, and the individual measures on gajw between summands for
m€[F n, F n+ 1) converge to geometric decay for almost all m as n→∞. While similar results
are known for many other recurrences, previous work focused on proving Gaussianity for the
number of summands or the average gap measure. We derive general conditions, which are
easily checked, that yield geometric decay in the individual gap measures of generalized
Zerkendorf decompositions attached to many linear recurrence relations.