Central Limit Theorems for Gaps of Generalized Zeckendorf Decompositions

Abstract

Zeckendorf proved that every integer can be written uniquely as a sum of non-adjacent Fibonacci numbers $\{1,2,3,5,\dots\}$. This has been extended to many other recurrence relations $\{G_n\}$ (with their own notion of a legal decomposition) and to proving that the distribution of the number of summands of an $M \in [G_n, G_{n+1})$ converges to a Gaussian as $n\to\infty$. We prove that for any non-negative integer $g$ the average number of gaps of size $g$ in many generalized Zeckendorf decompositions is $C_\mu n+d_\mu+o(1)$ for constants $C_\mu > 0$ and $d_\mu$ depending on $g$ and the recurrence, the variance of the number of gaps of size $g$ is similarly $C_\sigma n + d_\sigma + o(1)$ with $C_\sigma > 0$, and the number of gaps of size $g$ of an $M\in[G_n,G_{n+1})$ converges to a Gaussian as $n\to\infty$. The proof is by analysis of an associated two-dimensional recurrence; we prove a general result on when such behavior converges to a Gaussian, and additionally re-derive other results in the literature.