Auto-similarity in rational base number systems

TL;DR

This paper investigates the structure of integer representations in rational base systems, revealing a deep connection between minimal infinite words, their associated trees, and the topological properties of their spans.

Contribution

It introduces a novel analysis of the representation tree structure and minimal words in rational base systems, linking them with transducers and topological closure properties.

Findings

The minimal words form a highly non-regular tree structure.

A transducer can compute the minimal word for n+1 from n.

The topological closure of the set of spans is characterized.

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. 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 these two consecutive minimal words is the called the span of a node of the tree. The preceding construction allows to characterise the topological closure of the set of spans.