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.
Categories: math.NT