The Average Gap Distribution for Generalized Zeckendorf Decompositions
Abstract
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 nonadjacent Fibonacci numbers. This is now known as Zeckendorf's theorem [21], and similar decompositions exist for many other sequences ${G_{n+1} = c_1 G_{n} + ... + c_L G_{n+1L}}$ 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 $C_{\rm Lek} n$ for a nonzero constant; this was improved by others to show the number of summands has Gaussian fluctuations about this mean. Kolo$\breve{\rm g}$lu, Kopp, Miller and Wang [17, 18] 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 $m \in [G_n, G_{n+1})$ for special choices of the $c_i$'s. Specifically, we study the case where each $c_i \in {0,1}$ and there is a $g$ such that there are always exactly $g1$ zeros between two nonzero $c_i$'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 fardifference (i.e., signed) decompositions.
 Publication:

arXiv eprints
 Pub Date:
 August 2012
 DOI:
 10.48550/arXiv.1208.5820
 arXiv:
 arXiv:1208.5820
 Bibcode:
 2012arXiv1208.5820B
 Keywords:

 Mathematics  Number Theory;
 11B39;
 11B05 (Primary) 65Q30;
 60B10 (Secondary)
 EPrint:
 15 pages, version 2.1, final version (fixed a typo in a combinatorial identity)  to appear in the Fibonacci Quarterly