Time:

1:30pm, March. 2, 2018

Venue:

Room 236, East Building of Department of Chemistry, Faculty of Science(化学東館), University of Tokyo

Title:

Mysteries and marvels of rational base numeration systems

Jacques Sakarovitch
CNRS / Paris Diderot University  and  Telecom ParisTech


Abstract:
The definition of numeration systems with rational base, in a joint
work with S. Akiyama and Ch. Frougny (Israel J. Math., 2008),
has allowed to make some progress in a number theoretic problem,
by means of automata theory and combinatorics of words.
At the same time, it 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.

At first sight, these sets look rather chaotic and do not fit well
in the classical Chomsky hierarchy of languages. They all enjoy a
property that makes them defeat, so to speak, any kind of iteration
lemma. On the other hand, these sets also exhibit remarkable
regularity properties.

During the recent years, these regularities have been studied in a
series of joint papers with my student V. Marsault. In particular, we
have shown that periodic signatures are characteristic of the
representation languages in rational base numeration systems and
studied, jointly with S. Akiyama, a kind of autosimilarity property
that also leads to the construction of Cantor-like sets.

These languages still keep most of their mystery. The partial results
which will be presented call for further investigations on the subject
even stronger.