Optimal number representations in negative base

TL;DR

This paper investigates optimal number representations in negative and positive non-integer bases, deriving transformations for their generation and revealing that optimal representations are rare in negative bases, while reaffirming known results for positive bases.

Contribution

It derives the transformation for optimal representations in negative non-integer bases and shows their scarcity, providing new insights and alternative proofs for positive bases.

Findings

Almost no numbers have optimal representations in negative non-integer bases.

Derived the transformation generating optimal representations in negative bases.

Reaffirmed existing results for positive bases with an alternative proof.

Abstract

For a given base $\gamma$ and a digit set ${\mathcal B}$ we consider optimal representations of a number $x$, as defined by Dajani at al. in 2012. For a non-integer negative base $\gamma=-\beta<-1$ and the digit set ${\mathcal A}_\beta:={0,1,...,\lceil\beta\rceil-1}$ we derive the transformation which generates the optimal representation, if it exists. We show that -- unlike the case of negative integer base -- almost no $x$ has an optimal representation. For a positive base $\gamma=\beta>1$ and the alphabet ${\mathcal A}_\beta$ we provide an alternative proof of statements obtained by Dajani et al.