Summand minimality and asymptotic convergence of generalized Zeckendorf decompositions

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.