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 $n={\sum }_k{\epsilon }_k{U}_k$, where the digits ${\epsilon }_k$ are taken from a finite alphabet $\Sigma$ and $(U_k{)}_k$ is a linear recurrent sequence of Pisot type with $U_0=1$. The most prominent example of a base sequence $(U_k{)}_k$ is the sequence of Fibonacci numbers. We prove that the representations of minimal weight ${\sum }_k|{\epsilon }_k|$ 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 order of magnitude of the number of representations of a given integer to the joint spectral radius of a certain set of matrices.