Diophantische Approximationen

Abstract

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 $\mathbf{x}\in {\mathbb{Z}}^n$ 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. Wüstholz. 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 Wüstholz.

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 $g\ge 1$. 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 ${\theta }_1,\dots ,{\theta }_n$ with the measure of linear independence over $\mathbf{Q}$ of the numbers $1,{\theta }_1,\dots ,{\theta }_n$. M. Laurent introduced in his talk exponents which measure the sharping of the approximation to the point $\Theta =({\theta }_1,\dots ,{\theta }_n)$ by rational linear varieties of dimension $d$ for every integer $d,0\le d<n,$ and proved some inequalities connecting these exponents. The Khintchine's inequality follows as a special case. Another kind of transference ideas were used in the joint talk of V. Beresnevich and S. Vilani to state some metric Diophantine approximation results. The transference lemma in functional domain directed to applications in multiplicity estimates and algebraic independence theory was reported by P. Philippon.

The determination of the arithmetic nature of values of the Riemann zeta function $\zeta (s)$ at odd values $s\in \mathbf{Z}$, $s>3$, is one of the most challenging problems in number theory. After Apery's celebrated proof of the irrationality of $\zeta (3)$, it took over twenty years until T. Rivoal proved that there are infinitely many numbers among $\zeta (3),\zeta (5),\zeta (7),\dots$ that are linearly independent over $\mathbf{Q}$ and W. Zudilin stated that at least one of $\zeta (5),\zeta (7),\zeta (9),\zeta (11)$ is irrational number. The difficulties are connected to constructions of good rational approximations to the corresponding values of zeta function. All known constructions have a hypergeometric nature. In a joint talk C. Krattenthaler and T. Rivoal gave a survey of recent constructions and explained the proof of so called Denominator conjecture that is based on some identities between a very-well-poised hypergeometric series and a multiple sum due to G. Andrews. Some constructions of approximations to zeta-values with multiple real integrals were discussed in the talk of C. Viola. T. Rivoal presented a new proof of the irrationality of $\zeta (3)$ that uses the expansion of the Hurwitz zeta function in interpolation series of rational functions. Such an interpolation process was first studied by Rene Lagrange in 1935.

The arithmetic properties of values of the Tschakaloff function $T_q(z)$ have been investigated in many works. One of the open problems is to prove linear independence of values of $T_q(z)$ for rational $z$ with different values of the parameter $q$. In the joint talk of W. Zudilin and K. Väänänen some results of this kind were presented.

In 2005 C. Fuchs and A. Dujella gave a negative answer on the question of Euler about existence of four positive integers with the property that the product of any two of them plus sum of multipliers is a perfect square. In his lecture C. Fuchs discussed analogous question for any four integer numbers. The new result of A. Dujella, A. Filipin and C. Fuchs is the finiteness of the number of quadruples satisfying this condition. Moreover an effective bound for the size of the integers was proved. A. Dujella in his talk discussed another analogous problem: to find a set of positive distinct integers $S$ such that for any pair $x,y\in S$ the number $xy+1$ is a square. The set $S=\{1,3,8,120\}$ was found by Fermat. It is proved that $\#S\le 5$ and there exists not more than finitely many sets with 5 elements. But no example has ever been found. These problems are connected to lower bounds for linear forms in logarithms of algebraic numbers. Diophantine equations of another type were discussed in the talk of M. Bennett.

B. Adamczewski surveyed some results connected to some problem of Mahler and Mendés France involving tools from automata theory, combinatorics on words and Diophantine approximation.

Another excellent survey of results and open questions connected to Hilbert's tenth problem about universal algorithm for solution of Diophantine equations was given by Yu. Matiyasevich.