Redundancy of minimal weight expansions in Pisot bases

Motivated by multiplication algorithms based on redundant number representations, we study representations of an integer [Formula: see text] as a sum [Formula: see text] , where the digits [Formula: see text] are taken from a finite alphabet [Formula: see text] and [Formula: see text] is a linear recurrent sequence of Pisot type with [Formula: see text]. The most prominent example of a base sequence [Formula: see text] is the sequence of Fibonacci numbers. We prove that the representations of minimal weight [Formula: see text] 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.