A Gel'fond type criterion in degree two
A method in diophantine approximation (II)
A note on the weighted Khintchine-Groshev Theorem
Let denote the set of –approximable points in . The classical Khintchine–Groshev theorem assumes a monotonicity condition on the approximating functions . Removing monotonicity from the Khintchine–Groshev theorem is attributed to different authors for different cases of and . It can not be removed for as Duffin–Schaeffer provided the counter example. We deal with the only remaining case and thereby remove all unnecessary conditions from the Khintchine–Groshev theorem.
A note on two linear forms
We prove a result on approximations to a real number θ by algebraic numbers of degree ≤ 2 in the case when we have certain information about the uniform Diophantine exponent ω̂ for the linear form x₀ + θx₁ + θ²x₂.
A note on zero-one laws in metrical Diophantine approximation
A two-dimensional continued fraction algorithm with Lagrange and Dirichlet properties
A Lagrange Theorem in dimension 2 is proved in this paper, for a particular two dimensional continued fraction algorithm, with a very natural geometrical definition. Dirichlet type properties for the convergence of this algorithm are also proved. These properties proceed from a geometrical quality of the algorithm. The links between all these properties are studied. In relation with this algorithm, some references are given to the works of various authors, in the domain of multidimensional continued...
An effective Version of Faltings' Product Theorem.
An improvement of the quantitative subspace theorem
Application of Hermite-Mahler Polynomials to the Approximation of the Values of p-adic Exponential Function.
Approximation simultanée par des nombres algébriques
Nous étudions l'approximation simultanée de nombres complexes transcendants par des nombres algébriques de degré borné. Nous montrons que deux nombres qui ne sont pas simultanément bien approchables sont tous deux très bien approchables par des nombres algébriques de degré borné.
Approximation simultanée par des produits de puissances de nombres algébriques
Approximations simultanées de deux nombres réels
Approximations simultanées de deux séries formelles quadratiques
Around the Littlewood conjecture in Diophantine approximation
The Littlewood conjecture in Diophantine approximation claims thatholds for all real numbers and , where denotes the distance to the nearest integer. Its -adic analogue, formulated by de Mathan and Teulié in 2004, asserts thatholds for every real number and every prime number , where denotes the -adic absolute value normalized by . We survey the known results on these conjectures and highlight recent developments.
Asymptotisches Verhalten einer diophantischen Approximations-Funktion
Bad(s,t) is hyperplane absolute winning
J. An proved that for any s,t ≥ 0 such that s + t = 1, Bad (s,t) is (34√2)¯¹-winning for Schmidt's game. We show that using the main lemma from [An] one can derive a stronger result, namely that Bad (s,t) is hyperplane absolute winning in the sense of [BFKRW]. As a consequence, one can deduce the full Hausdorff dimension of Bad (s,t) intersected with certain fractals.
Best simultaneous diophantine approximations of Pisot numbers and Rauzy fractals
Best simultaneous diophantine approximations of some cubic algebraic numbers
Let be a real algebraic number of degree over whose conjugates are not real. There exists an unit of the ring of integer of for which it is possible to describe the set of all best approximation vectors of .’
Bounds for the solutions of Thue-Mahler equations and norm form equations
Conjecture de Littlewood et récurrences linéaires
Ce travail est essentiellement consacré à la construction d’exemples effectifs de couples de nombres réels à constantes de Markov finies, tels que et soient -linéairement indépendants, et satisfaisant à la conjecture de Littlewood.