Redundancy of minimal weight expansions in Pisot bases

TL;DR

This paper investigates minimal weight representations of integers in Pisot bases, showing they are recognized by finite automata and analyzing their asymptotic behavior and maximal order of representations.

Contribution

It proves that minimal weight representations in Pisot bases are recognized by finite automata and relates their maximal number to the joint spectral radius.

Findings

Minimal weight representations are recognized by finite automata.

An asymptotic formula for the average number of minimal weight representations.

The maximal order of the number of representations relates to the joint spectral radius.

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.