Optimal number representations in negative base

For a given base $γ$ 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 $γ=-β<-1$ and the digit set ${\mathcal A}_β:={0,1,...,\lceilβ\rceil-1}$ 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 ${\mathcal A}_β$ we provide an alternative proof of statements obtained by Dajani et al.

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