Optimal number representations in negative base

Abstract

For a given base γ and a digit set $\mathcal{B}$ we consider optimal representations of a number x, as defined by Dajani et al. [3]. For a non-integer negative base γ=−β<−1 and the digit set $\mathcal{A}_{\beta}:= \{0,1,\dots,\lceil\beta\rceil-1\}$ 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 γ=β>1 and the alphabet $\mathcal{A}_{\beta}$ we provide an alternative proof of statements obtained by Dajani et al.