Logic characterisation of p/q-recognisable sets

Let pq be a rational number. Numeration in base p q is defined by a function that evaluates each finite word over Ap = {0, 1, . . . , p− 1} to a rational number in some set Np q . In particular, Np q contains all integers and the literature on base pq usually focuses on the set of words that are evaluated to integers; it is a rather chaotic language which is not context-free. On the contrary, we study here the subsets of (Np q )d that are pq -recognisable, i.e. realised by finite automata over (Ap) . First, we give a characterisation of these sets as those definable in a first-order logic, similar to the one given by the Büchi-Bruyère Theorem for integer bases. Second, we show that the order relation and the modulo-q operator are not pq -recognisable.