Event Detail

Event Type: 
Department Colloquium
Date/Time: 
Monday, November 4, 2013 - 08:00 to 09:00
Location: 
Kidder 350

Speaker Info

Institution: 
TU Delft, the Netherlands
Abstract: 

  The familiar expansion of real numbers in terms of powers of $\beta = 10$ can be easily generalized to expansions to a non-integer base $\beta >1$.   A less obvious expansion for real numbers is as continued fractions, leading to best rational approximations in the appropriate sense.   

 It is fairly easy to randomize $\beta$-expansion in a meaningful way. Due to this, one can show that almost every irrational number $x$ has infinitely many expansions to the same base $\beta >1$. This has been used by Daubechies and her co-authors in applications to the so-called A/D-conversions.

 It is not immediately clear how other expansions, in particular continued fraction expansions, can be randomized in a natural way. However, recent work by Maxwell Anselm and Steven Weintraub show a way to do this. In applying this,  we find many interesting and new continued fraction algorithms, yielding many new questions.

This is joint work with Karma Dajani (Utrecht) and Niels van der Wekken (Delft).