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 $n=\sum_k ε_k U_k$, where the digits $ε_k$ are taken from a finite alphabet $Σ$ 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 $\sum_k|ε_k|$ 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.