Redundancy of minimal weight expansions in Pisot bases

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.