The Distribution of Gaps between Summands in Generalized Zeckendorf Decompositions

TL;DR

This paper investigates the distribution of gaps between summands in generalized Zeckendorf decompositions, revealing geometric decay of large gaps, recurrence-dependent small gap distributions, and a log-scale distribution for the longest gap, applicable both on average and almost surely.

Contribution

It extends Zeckendorf's theorem to general linear recurrences, analyzing gap distributions and longest gaps, with explicit decay rates and recurrence-dependent distributions, both on average and for individual cases.

Findings

Large gaps decay geometrically in probability.

Small gap distribution depends on recurrence coefficients.

Longest gap distribution is double exponential and log-scaled.

Abstract

Zeckendorf proved that any integer can be decomposed uniquely as a sum of non-adjacent Fibonacci numbers, $F_n$. Using continued fractions, Lekkerkerker proved the average number of summands of an $m \in [F_n, F_{n+1})$ is essentially $n/(\varphi^2 +1)$, with $\varphi$ the golden ratio. Miller-Wang generalized this by adopting a combinatorial perspective, proving that for any positive linear recurrence the number of summands in decompositions for integers in $[G_n, G_{n+1})$ converges to a Gaussian distribution. We prove the probability of a gap larger than the recurrence length converges to decaying geometrically, and that the distribution of the smaller gaps depends in a computable way on the coefficients of the recurrence. These results hold both for the average over all $m \in [G_n, G_{n+1})$, as well as holding almost surely for the gap measure associated to individual $m$. The techniques can also be used to determine the distribution of the longest gap between summands, which we prove is similar to the distribution of the longest gap between heads in tosses of a biased coin. It is a double exponential strongly concentrated about the mean, and is on the order of $\log n$ with computable constants depending on the recurrence.