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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.

On the multiples of a badly approximable vector

Yann Bugeaud — 2015

Acta Arithmetica

Let d be a positive integer and α a real algebraic number of degree d + 1. Set α ̲ : = ( α , α ² , . . . , α d ) . It is well-known that c ( α ̲ ) : = l i m i n f q q 1 / d · | | q α ̲ | | > 0 , where ||·|| denotes the distance to the nearest integer. Furthermore, c ( α ̲ ) n - 1 / d c ( n α ̲ ) n c ( α ̲ ) for any integer n ≥ 1. Our main result asserts that there exists a real number C, depending only on α, such that c ( n α ̲ ) C n - 1 / d for any integer n ≥ 1.

On the rational approximation to the Thue–Morse–Mahler numbers

Yann Bugeaud — 2011

Annales de l’institut Fourier

Let ( t k ) k 0 be the Thue–Morse sequence on { 0 , 1 } defined by t 0 = 0 , t 2 k = t k and t 2 k + 1 = 1 - t k for k 0 . Let b 2 be an integer. We establish that the irrationality exponent of the Thue–Morse–Mahler number k 0 t k b - k is equal to 2 .

On simultaneous rational approximation to a real number and its integral powers

Yann Bugeaud — 2010

Annales de l’institut Fourier

For a positive integer n and a real number ξ , let λ n ( ξ ) denote the supremum of the real numbers λ such that there are arbitrarily large positive integers q such that | | q ξ | | , | | q ξ 2 | | , ... , | | q ξ n | | are all less than q - λ . Here, | | · | | denotes the distance to the nearest integer. We study the set of values taken by the function λ n and, more generally, we are concerned with the joint spectrum of ( λ 1 , ... , λ n , ... ) . We further address several open problems.

Quantitative versions of the Subspace Theorem and applications

Yann Bugeaud — 2011

Journal de Théorie des Nombres de Bordeaux

During the last decade, several quite unexpected applications of the Schmidt Subspace Theorem were found. We survey some of these, with a special emphasize on the consequences of quantitative statements of this theorem, in particular regarding transcendence questions.

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.

Extensions of the Cugiani-Mahler theorem

Yann Bugeaud — 2007

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In 1955, Roth established that if ξ is an irrational number such that there are a positive real number ε and infinitely many rational numbers p / q with q 1 and | ξ - p / q | < q - 2 - ε , then ξ is transcendental. A few years later, Cugiani obtained the same conclusion with ε replaced by a function q ε ( q ) that decreases very slowly to zero, provided that the sequence of rational solutions to | ξ - p / q | < q - 2 - ε ( q ) is sufficiently dense, in a suitable sense. We give an alternative, and much simpler, proof of Cugiani’s Theorem and extend it to simultaneous...

Automatic continued fractions are transcendental or quadratic

Yann Bugeaud — 2013

Annales scientifiques de l'École Normale Supérieure

We establish new combinatorial transcendence criteria for continued fraction expansions. Let  α = [ 0 ; a 1 , a 2 , ... ] be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients ( a ) 1 of  α is not ‘too simple’ (in a suitable sense) and cannot be generated by a finite automaton.

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