The Average Gap Distribution for Generalized Zeckendorf Decompositions

TL;DR

This paper investigates the distribution of gaps between summands in generalized Zeckendorf decompositions, revealing that gaps shorter than a certain length do not occur and longer gaps decay geometrically, with results applicable to various recurrence-based sequences.

Contribution

It introduces a combinatorial approach to analyze gap distributions in generalized Zeckendorf decompositions, extending understanding beyond previous average summand results.

Findings

Gaps shorter than a certain length do not occur.

Probability of larger gaps decays geometrically.

Results apply to sequences like Fibonacci and Tribonacci.

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 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 $C_{{\rm Lek}} n$ for a non-zero 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 $g-1$ 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.