# Oberwolfach Reports

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**Volume 4, Issue 2, 2007, pp. 1115–1190**

**DOI: 10.4171/OWR/2007/21**

Published online: 2008-03-31

Diophantische Approximationen

Yuri V. Nesterenko^{[1]}and Peter Schlickewei

^{[2]}(1) Moscow Lomonosov State University, Russian Federation

(2) Philipps-Universität, Marburg, Germany

The workshop Diophantische Approximationen (Diophantine
approximations), organised by Yuri V. Nesterenko (Moscow) and
Hans-Peter Schlickewei (Marburg) was held April 15th - April 21st,
2007. This meeting was well attended with over 40 participants with
broad geographic representation. This workshop was a nice blend of
researchers with various backgrounds. All the participants were
inspired by the fact that the conference immediately followed the
300 anniversary of Euler birth (15.04.1707).
Loosely speaking Diophantine approximation is a branch of Number
Theory that can be described as a study of the solvability of
inequalities in integers, though this main theme of the subject is
often unbelievably generalized. As an example, one can be interested
in properties of rational points of algebraic varieties defined over
an algebraic number field. The conference was concerned with a
variety of problems of this kind. Below we briefly recall some of
the results presented at this conference, thus outlining some modern
lines of investigation in Diophantine approximation. More details
can be found in the corresponding abstracts.
The classical Subspace Theorem claims that all integer solutions
${\bf x}\in \mathbb{Z}^n$ of a special system of linear inequalities
with algebraic coefficients belong to a finite number of linear
subspaces of $\mathbb{R}^n$. This theorem proved by W.Schmidt in
70-th of 20-th century is a far reaching generalization of the
famous theorem of Roth about approximation of algebraic numbers by
rationals. Subsequently Schmidt gave an estimate for the number of
such subspaces. This result was improved and extended by
H.P.~Schlickewei and J.H.~Evertse. Another approach to the proof of
Schmidt's theorem was proposed by G.~Faltings and G.~W\"ustholz. In
the joint talk of J.H.~Evertse and R.~Ferretti the upper bound for
the number of the subspaces in question was significantly improved
by combining ideas of Schmidt, Faltings and W\"ustholz.
Results of this kind have many applications. For example Y.~Bugeaud
in his talk announced joint with J.H.~Evertse theorem that for any
real algebraic number $\xi$ and any integer $b>1$ the number of
distinct blocks of $n$ letters occurring in the $b$-ary expansion of
$\xi$ asymptotically exceed $n(\log n)^\eta$ for any positive
$\eta<1/14$. Another example is connected to the classical theorem
of Siegel about integer points on curves of genus $g\geq 1$. In the
survey talk of Yu.~Bilu another proof of this theorem based on
quantitative version of Subspace Theorem was presented. This proof
belongs to P.~Corvaja and U.~Zannier (2002) who applied their
arguments to integral points on surfaces. Corresponding results were
presented in the talk of Bilu the same as more precise statement of
A.~Levin and P.~Autissier.
Talks of P.~Habegger, A.~Galateau were devoted to the problem of
lower bounds of heights on subvarieties of group varieties that is
analogous to the classical Lehmer problem. Earlier works in this
direction belong to E.~Bombieri, D.~Masser, U.~Zannier, F.~Amoroso,
S.~David and P.~Philippon.
P.~Mihailescu discussed in his talk so called Fermat-Catalan
equation. In particular he gave some sufficient conditions on prime
numbers $p, q$ providing existence only trivial rational solutions
for the equation $x^p+y^q=1$. The methods used by Mihailescu have a
cyclotomic nature and they combine class field conditions with some
new approximation techniques.
The well-known Khintchine Transference Principle relates the measure
of simultaneous rational approximation of the real numbers
$\theta_1,\ldots ,\theta_n$ with the measure of linear independence
over $\bf Q$ of the numbers $1, \theta_1,\ldots,\theta_n$.
M.~Laurent introduced in his talk exponents which measure the
sharping of the approximation to the point $\Theta=(\theta_1,\ldots
,\theta_n)$ by rational linear varieties of dimension $d$ for every
integer $d, 0\leq d

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Nesterenko Yuri, Schlickewei Peter: Diophantische Approximationen. *Oberwolfach Rep.* 4 (2007), 1115-1190. doi: 10.4171/OWR/2007/21