Optimal number representations in negative base

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.

9 pages