The theory of Diophantine approximation is a classical topic in number theory and deals with approximation
properties of real numbers by rational (or algebraic) ones. During the last years it has become a very active field in
number theory with applications to random numbers and Diophantine equations. There are also many interesting
open und unsolved problems.
In this project we plan to investigage three celebrated questions in Diophantine approximation. A first one consists
in studying approximation properties of real numbers with `special` continued fraction expansions. Many progress
have been made recently on this topic: for instance, such numbers have given counterexamples to widely believed
conjectures in Diophantine approximation. A second one can be formulated as follows: is the sequence of decimal
parts of the powers of a rational number a `well` distributed sequence, i.e., does it behave like a random sequence?
A third one, called the Littlewood Conjecture, concerns simultaneous approximation of pairs of real numbers by
rational number with the same denominator. The second two problems are reputed to be difficult and very little is
known towards their resolution, although there has been some recent progress. We plan to develop new methods
yielding partial results, for example metric results on the Hausdorff dimension. There will be other appllications,
too, e.g. for the study of the discrepancy of special sequences.