Auto-similarity in rational base number systems

Abstract

This work is a contribution to the study of set of the representations of integers in a rational base number system. This prefix-closed subset of the free monoid is naturally represented as a highly non regular tree whose nodes are the integers and whose subtrees are all distinct. With every node of that tree is then associated a minimal infinite word (and a maximal infinite word). The main result is that a sequential transducer which computes for all n the minimal word associated with n + 1 from the one associated with n, has essentially the same underlying graph as the tree itself. These infinite words are then interpreted as representations of real numbers; the difference between the numbers represented by the maximal and minimal word associated with n is called the span of n. The preceding construction allows to characterise the topological closure of the set of spans. © 2013 Springer-Verlag Berlin Heidelberg.