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A note on the weighted Khintchine-Groshev Theorem

Mumtaz Hussain, Tatiana Yusupova (2014)

Journal de Théorie des Nombres de Bordeaux

Let W ( m , n ; ψ ̲ ) denote the set of ψ 1 , ... , ψ n –approximable points in ℝ m n . The classical Khintchine–Groshev theorem assumes a monotonicity condition on the approximating functions ψ ̲ . Removing monotonicity from the Khintchine–Groshev theorem is attributed to different authors for different cases of m and n . It can not be removed for m = n = 1 as Duffin–Schaeffer provided the counter example. We deal with the only remaining case m = 2 and thereby remove all unnecessary conditions from the Khintchine–Groshev theorem.

A note on two linear forms

Nikolay Moshchevitin (2014)

Acta Arithmetica

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 two-dimensional continued fraction algorithm with Lagrange and Dirichlet properties

Christian Drouin (2014)

Journal de Théorie des Nombres de Bordeaux

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...

Approximation simultanée par des nombres algébriques

Yann Bugeaud (2003)

Journal de théorie des nombres de Bordeaux

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.

Around the Littlewood conjecture in Diophantine approximation

Yann Bugeaud (2014)

Publications mathématiques de Besançon

The Littlewood conjecture in Diophantine approximation claims that inf q ≥ 1 q · ∥ q α ∥ · ∥ q β ∥ = 0 holds for all real numbers α and β , where ∥ · ∥ denotes the distance to the nearest integer. Its p -adic analogue, formulated by de Mathan and Teulié in 2004, asserts that inf q ≥ 1 q · ∥ q α ∥ · | q | p = 0 holds for every real number α and every prime number p , where | · | p denotes the p -adic absolute value normalized by | p | p = p - 1 . We survey the known results on these conjectures and highlight recent developments.

Bad(s,t) is hyperplane absolute winning

Erez Nesharim, David Simmons (2014)

Acta Arithmetica

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 some cubic algebraic numbers

Nicolas Chevallier (2002)

Journal de théorie des nombres de Bordeaux

Let α be a real algebraic number of degree 3 over ℚ whose conjugates are not real. There exists an unit ζ of the ring of integer of K = ℚ ( α ) for which it is possible to describe the set of all best approximation vectors of θ = ( ζ , ζ 2 ) .’

Conjecture de Littlewood et récurrences linéaires

Bernard de Mathan (2003)

Journal de théorie des nombres de Bordeaux

Ce travail est essentiellement consacré à la construction d’exemples effectifs de couples ( α , β ) de nombres réels à constantes de Markov finies, tels que 1 , α et β soient 𝐙 -linéairement indépendants, et satisfaisant à la conjecture de Littlewood.

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