# The Successor Function for Minimal Words in Rational Base Numeration Systems

- 11:00 14th June 2016 ( week 8, Trinity Term 2016 )Room 441, Wolfson Building

Rational base numeration systems were introduced by Akiyama, Frougny and

Sakarovitch in 2008 and allowed to make some progress in a number theoretic

problem, by means of automata theory and combinatorics of words. At the

same time, they raised the problem of understanding the structure of the

sets of the representations of the integers in these systems from the point

of view of formal language theory.

We present here a contribution to the study of this language. Since it is

prefix-closed, it is naturally represented as an highly non-regular

(infinite) tree whose nodes are the integers and whose subtrees are

pairwise distinct. With every node of that tree is then associated a

minimal infinite word and we study the function that computes for all n the

minimal word associated with (n+1) from the one associated with n. The

main result is that this function is realised by an infinite, sequential

and letter-to-letter transducer which 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.