title: Redundancy of minimal weight expansions in Pisot bases.
abstract: Motivated by multiplication algorithms based on redundant number representations, we study representations of an integer n as a sum n=∑kεkUk, where the digits εk are taken from a finite alphabet Σ and (Uk)k is a linear recurrent sequence of Pisot type with U0=1. The most prominent example of a base sequence (Uk)k is the sequence of Fibonacci numbers. We prove that the representations of minimal weight ∑k|εk| are recognised by a finite automaton and obtain an asymptotic formula for the average number of representations of minimal weight. Furthermore, we relate the maximal number of representations of a given integer to the joint spectral radius of a certain set of matrices.
authors.affiliation: Institut für Analysis und Computational Number Theory, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria.
authors.forename: Peter J
authors.initials: PJ
authors.lastname: Grabner
authors.forename: Wolfgang
authors.initials: W
authors.lastname: Steiner