Math.Koll. am 25.04.07/Prof. Sakarovich

[Institut für Mathematik]

Mathematisches Kolloquium

EINLADUNG

zu einem

VORTRAG

von

Prof. Jacques Sakarovich (LTCI, ENST/CNRS, Paris)

mit dem Thema: ''Powers of rationals modulo 1 and rational base number
systems''

Abstract:
In a first part, I'll present the framework of this work, the relationships
between the writing of numbers and finite automata theory. They begin with
toy examples of automata (due to Blaise Pascal indeed) and more seriously
with the beautiful Cobham's theorem (1969). The study of non standard
numeration systems has made these links even tighter. Representation in
integer base with signed digits was popularized in computer arithmetic by
Avizienis and clearly uses finite automata. When the base is a real number
$\beta > 1$, a number can be given several representation (even on the
canonical alphabet) and the normal one, that is the one obtained by the
greedy algorithm can be computed (from the other representations) by a
finite automaton if and only if $\beta$ is a Pisot number, that is an
algebraic integer such that all its Galois conjugates have a modulus smaller
than $1$. In a second part, I would like to present some recent results on
rational base systems obtained in cooperation with my colleagues S. Akiyama
(Niigata Univ.)and Ch. Frougny (Paris 8 Univ.). A new method for
representing positive integers and real numbers in a rational base is
considered. It amounts to computing the digits from right to left, least
significant first. Every integer has a unique such expansion. In contrast
with the Pisot case, the set of expansions of the integers is not a regular
language but nevertheless addition can be performed by a letter-to-letter
finite right transducer. Every real number has at least one such expansion
and a countable infinite set of them have more than one. We explain how
these expansions can be approximated and characterize the expansions of
reals that have two expansions. These results are developped not only for
their own sake but also as they relate to other problems in combinatorics
and number theory. A first example is a new interpretation and expansion of
the constant $K(p)$ from the so-called ``Josephus problem''. More important,
these expansions in the base~$p/q$ allow us to make some progress in the
problem of the distribution of the fractional part of the powers of rational
numbers.


Zeit: Mittwoch, 25. April 2007, 16.00 Uhr (Kaffeejause), anschlieszend 16.30
Uhr Vortrag

Ort: Fakultaet fuer Mathematik der Universitaet Wien, Nordbergstr. 15,
Seminarraum C 2.09

Harald Rindler
Karl Auinger