18,820 research outputs found
Report on some recent advances in Diophantine approximation
A basic question of Diophantine approximation, which is the first issue we
discuss, is to investigate the rational approximations to a single real number.
Next, we consider the algebraic or polynomial approximations to a single
complex number, as well as the simultaneous approximation of powers of a real
number by rational numbers with the same denominator. Finally we study
generalisations of these questions to higher dimensions. Several recent
advances have been made by B. Adamczewski, Y. Bugeaud, S. Fischler, M. Laurent,
T. Rivoal, D. Roy and W.M. Schmidt, among others. We review some of these
works.Comment: to be published by Springer Verlag, Special volume in honor of Serge
Lang, ed. Dorian Goldfeld, Jay Jorgensen, Dinakar Ramakrishnan, Ken Ribet and
John Tat
Thue's Fundamentaltheorem, I: The General Case
In this paper, Thue's Fundamentaltheorem is analysed. We show that it
includes, and often strengthens, known effective irrationality measures
obtained via the so-called hypergeometric method as well as showing that it can
be applied to previously unconsidered families of algebraic numbers.
Furthermore, we extend the method to also cover approximation by algebraic
numbers in imaginary quadratic number fields.Comment: accepted version (Acta Arithmetica
Arithmetic theory of E-operators
In [S\'eries Gevrey de type arithm\'etique I Th\'eor\'emes de puret\'e et de
dualit\'e, Annals of Math. 151 (2000), 705--740], Andr\'e has introduced
E-operators, a class of differential operators intimately related to
E-functions, and constructed local bases of solutions for these operators. In
this paper we investigate the arithmetical nature of connexion constants of
E-operators at finite distance, and of Stokes constants at infinity. We prove
that they involve values at algebraic points of E-functions in the former case,
and in the latter one, values of G-functions and of derivatives of the Gamma
function at rational points in a very precise way. As an application, we define
and study a class of numbers having certain algebraic approximations defined in
terms of E-functions. These types of approximations are motivated by the
convergents to the number e, as well as by recent constructions of
approximations to Euler's constant and values of the Gamma function. Our
results and methods are completely different from those in our paper [On the
values of G-functions, Commentarii Math. Helv., to appear], where we have
studied similar questions for G-functions
Effective Irrationality Measures and Approximation by Algebraic Conjugates
In this paper, we present a result on using algebraic conjugates to form a
sequence of approximations to an algebraic number, and in this way obtain
effective irrationality measures for related algebraic numbers. From this
result, we are able to generalise Thue's Fundamentaltheorem.Comment: 14 page
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