THE AVERAGE GAP DISTRIBUTION FOR GENERALIZED ZECKENDORF DECOMPOSITIONS

An interesting characterization of the Fibonacci numbers is that if we write them as F-1 = 1, F-2 = 2, F-3 = 3, F-4 = 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 {G(n+1) = c(1) G(n) + . . . + c(L)G(n+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 [G(n), G(n+1)) is on the order of C(Lek)n for a non-zero constant; this was improved by others to show the number of summands has Gaussian fluctuations about this mean. Kologlu, 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 is an element of [G(n), G(n+1)) for special choices of the c(i)'s. Specifically, we study the case where each c(i) is an element of{0, 1} and there is a g such that there are always exactly g - 1 zeros between two non-zero c(i)'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.