Summand minimality and asymptotic convergence of generalized Zeckendorf decompositions

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 \(c_1, c_t \ge 1\), 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 \(n \rightarrow \infty \); 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.