Greedy expansions and sets with deleted digits

We generalize a result of Dar{\'o}czy and K{\'a}tai, on the characterization of univoque numbers with respect to a non-integer base \cite{DarKat95} by relaxing the digit alphabet to a generic set of real numbers. We apply the result to derive the construction of a B\"uchi automaton accepting all and only the greedy sequences for a given base and digit set. In the appendix we prove a more general version of the fact that the expansion of an element $x\in \QQ(q)$ is ultimately periodic, if $q$ is a Pisot number.