BGU SEMINAR IN PROBABILITY AND ERGODIC THEORY
 
Shalom all,
 
SPEAKER: Menny Aka, Hebrew University
 
DATE: 19.12.10
 
TIME:  14:00 
 
PLACE: Seminar room -101
 
TITLE: On the Continued fraction expansion of quadratic irrationals
 
ABSTRACT: It is well known that quadratic irrationals are characterized as
the numbers whose continued fraction expansion is eventually periodic.
Each quadratic irrational exhibits then a certain statistics in its period
(for example, one can measure the frequency of the digit 1 in the period).
 
I will present a recent result regarding the continued fraction
expansionsof certain sequences of quadratic irrationals.  We prove the
following Theorem:
Let x be a quadratic irrational and p a prime. Then the statistics of the
period of the continued fraction expansion of p^nx converges to the "right"
statistics; i.e. to the one given by the
Gauss-Kuzmin measure.
Our proof exploits a classical connection between the continued fraction
expansion and dynamics on the modular surface. I will review this connection
and explain the proof of an equivalent equidistribution result  on a
S-arithmetic homogeneous space.
 
This is joint work with U. Shapira.
 
---------------------------------------------------------
Technion Math Net-2 (TECHMATH2)
Editor: Michael Cwikel   <techm@math.technion.ac.il> 
Announcement from: Barak Weiss   <barakw@cs.bgu.ac.il>