The Average Gap Distribution For Generalized Zeckendorf Decompositions

Abstract

An interesting characterization of the Fibonacci numbers is that if we write them as F1 = 1, F2 = 2, F3 = 3, F4 = 5, …, then every positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. This is now known as Zeckendorf’s Theorem [21], and similar decompositions exist for many other sequences {Gn+1 = c1Gn +· · ·+cLGn+1-L} arising from recurrence relations. Much more is known. Using continued fraction approaches, Lekkerkerker [15] proved the average number of summands needed for integers in [Gn,Gn+1) is on the order of CLekn for a non-zero constant; this was improved by others to show the number of summands has Gaussian fluctuations about this mean.Koloğlu, Kopp, Miller and Wang [13, 17] recently recast the problem combinatorially, reproving and generalizing these results. We use this new perspective to investigate the distribution of gaps between summands. We explore the average behavior over all m ∈ [Gn,Gn+1) for special choices of the ci’s. Specifically, we study the case where each ci ∈ {0, 1} and there is a g such that there are always exactly g-1 zeros between two non-zero ci’s; note this includes the Fibonacci, Tribonacci and many other important special cases. We prove there are no gaps of length less than g, and the probability of a gap of length j > g decays geometrically, with the decay ratio equal to the largest root of the recurrence relation. These methods are combinatorial and apply to related problems; we end with a discussion of similar results for far-difference (i.e., signed) decompositions.