Summand minimality and asymptotic convergence of generalized Zeckendorf decompositions

TL;DR

This paper investigates the properties of generalized Zeckendorf decompositions (gzd), proving conditions for summand minimality, developing an algorithm for conversion, and analyzing the asymptotic convergence of gzds.

Contribution

It establishes that gzd is summand minimal if and only if the signature sequence is weakly decreasing, and analyzes the convergence behavior of gzds for large indices.

Findings

gzd uses the fewest summands when the signature is weakly decreasing

An algorithm for converting any representation to gzd is developed and analyzed

gzds of certain linear combinations converge with three distinct behaviors

Abstract

Given a recurrence sequence $H$, with $H_n = c_1 H_{n-1} + \dots + c_t H_{n-t}$ where $c_i \in \mathbb{N}_0$ for all $i$ and $c_1, c_t \geq 1$, the generalized Zeckendorf decomposition (gzd) of $m \in \mathbb{N}_0$ is the unique representation of $m$ using $H$ composed of blocks lexicographically less than $\sigma = (c_1, \dots, c_t)$. We prove that the gzd of $m$ uses the fewest number of summands among all representations of $m$ using $H$, for all $m$, if and only if $\sigma$ is weakly decreasing. We develop an algorithm for moving from any representation of $m$ to the gzd, the analysis of which proves that $\sigma$ weakly decreasing implies summand minimality. We prove that the gzds of numbers of the form $v_0 H_n + \dots + v_\ell H_{n-\ell}$ converge in a suitable sense as $n \to \infty$, furthermore we classify three distinct behaviors for this convergence. We use this result, together with the irreducibility of certain families of polynomials, to exhibit a representation with fewer summands than the gzd if $\sigma$ is not weakly decreasing.