(重複で受け取られた方はご容赦ください) 皆様，

秋田大学の新屋です．

以下の要領で3/2(金)に東大にてJacques Sakarovitch先生の講演会が開催されます．

なお，前日(2/28~3/1)の東工大でのSakarovitch先生の連続講演会とは講演内容が別の 新規なものとなっております．どうぞふるってご参加ください．

秋田大学 数理科学コース 新屋良磨 [email protected] mailto:[email protected]

========== 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. ==========