Abstract:
After a brief introduction to basic problems and concepts of Diophantine
approximation in a broad sense, the talk will specialize on the dependent
variable theory, or equivalently restricting the vectors to be approximated to
manifolds in Euclidean spaces of arbitrary dimension. Particular emphasis
will be on a special class of intensely studied curves, called Veronese curves.
Their investigation is in particular suggested by their close connection to a
famous open problem by Wirsing dating back to 1961. Roughly speaking, it
asks whether the classical Theorem of Dirichlet on approximation of real
numbers by rational numbers can be generalized to the setting of
approximation by algebraic numbers of higher degree. A major result from
the habilitation thesis (jointly with D. Badziahin) claims a significantly
improved lower bound for the associated exponent of approximation defined
by Wirsing himself. We state this result, and then go on to introduce some
more classical exponents of Diophantine approximation going back to K.
Mahler and Bugeaud & Laurent, partly closely related to Wirsing’s problem
again, but also of independent interest. We present selected results on these
exponents, some of them due to the presenter. This includes equivalent
variants of Mahler’s classification of transcendental real numbers into the
classes S, T, U, in terms of (sequences of) other exponents of approximation
than the one used by Mahler.