Limiting Distributions in Generalized Zeckendorf Decompositions

TL;DR

This paper explores generalized Zeckendorf decompositions with variable bin sizes and restrictions, establishing conditions for unique representations and Gaussian distribution of summands using the Lyapunov CLT.

Contribution

It extends Zeckendorf's theorem to more general sequences with variable bin sizes and restrictions, providing new conditions for uniqueness and distribution convergence.

Findings

Sequences have unique decompositions under specified conditions.

Number of summands converges to a Gaussian distribution.

Main tool used is the Lyapunov Central Limit Theorem.

Abstract

An equivalent definition of the Fibonacci numbers is that they are the unique sequence such that every integer can be written uniquely as a sum of non-adjacent terms. We can view this as we have bins of length 1, we can take at most one element from a bin, and if we choose an element from a bin we cannot take one from a neighboring bin. We generalize to allowing bins of varying length and restrictions as to how many elements may be used in a decomposition. We derive conditions on when the resulting sequences have uniqueness of decomposition, and (similar to the Fibonacci case) when the number of summands converges to a Gaussian; the main tool in the proofs here is the Lyaponuv Central Limit Theorem.