Central Limit Theorems for Gaps of Generalized Zeckendorf Decompositions

Ray Li; Steven J. Miller

paper

Central Limit Theorems for Gaps of Generalized Zeckendorf Decompositions

arXiv:1606.08110

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 $C_{μ} n + d_{μ} + o (1)$ for constants $C_{μ} > 0$ and $d_{μ}$ depending on $g$ and the recurrence, the variance of the number of gaps of size $g$ is similarly $C_{σ} n + d_{σ} + o (1)$ with $C_{σ} > 0$ , 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.

arXiv commentVersion 1.2, 32 pages (fixed a few typos, added a reference)

Central Limit Theorems for Gaps of Generalized Zeckendorf Decompositions

Abstract

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