On subtrees of the representation tree in rational base numeration systems

Every rational number p/q defines a rational base numeration system in whichevery integer has a unique finite representation, up to leading zeroes. Thiswork is a contribution to the study of the set of the representations ofintegers. This prefix-closed subset of the free monoid is naturally representedas a highly non-regular tree. Its nodes are the integers, its edges bear labelstaken in {0,1,...,p-1}, and its subtrees are all distinct. We associate with each subtree (or with its root n) three infinite words. Thebottom word of n is the lexicographically smallest word that is the label of abranch of the subtree. The top word of n is defined similarly. The span-word ofn is the digitwise difference between the latter and the former. First, we show that the set of all the span-words is accepted by an infiniteautomaton whose underlying graph is essentially the same as the tree itself.Second, we study the function that computes for all n the bottom wordassociated with n+1 from the one associated with n, and show that it isrealised by an infinite sequential transducer whose underlying graph is onceagain essentially the same as the tree itself. An infinite word may be interpreted as an expansion in base p/q after theradix point, hence evaluated to a real number. If T is a subtree whose root isn, then the evaluations of the labels of the branches of T form an interval of$\mathbb{R}$. The length of this interval is called the span of n and is equalto the evaluation of the span-word of n. The set of all spans is then a subsetof R and we use the preceding construction to study its topological closure. Weshow that it is an interval when p is greater than or equal to 2q-1, and aCantor set of measure zero otherwise.Comment: 33 page