event
Mardi 5 juin 2007
Mardi 5 juin 2007
place
IRMA
IRMA
Organisateur : Yann Bugeaud
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Mardi 5 juin 2007
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10:30 - 11:00
des participants au salon de l'IRMA (café). -
11:00 - 11:45
A. Dubickas, Vilnius
On the powers of a number modulo 1.
We shall review on some old problems and sketch few recent results concerning the distribution of the sequence ab^n modulo 1, where $a \not= 0$ and b > 1 are two fixed real numbers and n runs through every non-negative integer. -
14:00 - 14:45
Ph. Habegger, Bâle
Factorizing certain trinomials over cyclotomic fields with applications to heights.
Trinomials such as $X^n\pm X^m\pm 1$ were factorized over the rationals by Ljunggren and Tvergberg. Later, Schinzel did the same over the maximal abelian extension of the ratio- nals. Over this field we present a factorization result where the coefficients of the trinomial are allowed to be any root of unity and n and m are coprime. In particular we show that the resulting trinomials are usually irreducible. We also present several applications of this factorization result in the context of multiplicatively dependent solutions of x + y = 1. -
15:00 - 15:45
C. Fuchs, Zurich
A problem of Diophantus and its variants.
One of the problems Diophantus of Alexandria was interested in was to find sets with the property that the product of any two of its distinct elements increased by one is a perfect square. There has been a lot of activity towards understanding such sets in the recent years, most notably by A. Dujella who showed that there exist at most finitely many sets consisting of five integers with the property from above, the conjecture being that there exists no such set at all. I will give an overview over some of the known results related to this problem and discuss variants of it, some of which were also already investigated by Diophantus, Fermat and Euler, and I will outline which methods are used to attack the proof of such problems. -
15:45 - 16:15
café -
16:15 - 17:00
S. Kristensen, Arhus
Metrical results for ful ly non-linear Diophantine approximation.
Diophantine approximation of a single linear form is concerned with inequalities of the kind $$ |q_1 x_1 + \cdots + q_n x_n - p| < \psi(\max\{|q_i|\}), $$ where the qi and the p are integers and ? is some function. The metrical theory studies the measure theoretical structure of the set of real vectors x for which this inequality has infinitely many integer solutions q1 , . . . , qn , p. When we restrict the solutions q1 , . . . , qn , p to lie in certain subsets of the integers, this problem becomes more difficult. While classical methods can be used when p is unrestricted, these are no longer sufficient for a restricted set of p’s.
A recent result gives a complete metrical description of the set of such vectors when the qi and p are all restricted to being perfect squares and ? is assumed to be decreasing. The problem is related to the solubility of an inhomogeneous wave equation on a torus. We will outline the proof of this result and give some possible extensions. This is joint work with Beresnevich, Dodson and Levesley.