Central Limit Theorems for Gaps of Generalized Zeckendorf Decompositions

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 $n\to \mathrm{\infty }$. 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$,andthenumberofgapsofsize$g$ofan$M\in[G_n,G_{n+1})$convergestoaGaussianas$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.