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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.

Approximations diophantiennes des nombres sturmiens

Martine Queffélec (2002)

Journal de théorie des nombres de Bordeaux

Nous établissons pour tout nombre sturmien (de développement dyadique sturmien) des propriétés d'approximation diophantienne très précises, ne dépendant que de l'angle de la suite sturmienne, généralisant ainsi des travaux antérieurs de Ferenczi-Mauduit et Bullett-Sentenac.

Continued fraction expansions for complex numbers-a general approach

S. G. Dani (2015)

Acta Arithmetica

We introduce a general framework for studying continued fraction expansions for complex numbers, and establish some results on the convergence of the corresponding sequence of convergents. For continued fraction expansions with partial quotients in a discrete subring of ℂ an analogue of the classical Lagrange theorem, characterising quadratic surds as numbers with eventually periodic continued fraction expansions, is proved. Monotonicity and exponential growth are established for the absolute values...

Diophantine approximations with Fibonacci numbers

Victoria Zhuravleva (2013)

Journal de Théorie des Nombres de Bordeaux

Let F n be the n -th Fibonacci number. Put ϕ = 1 + 5 2 . We prove that the following inequalities hold for any real α :1) inf n | | F n α | | ϕ - 1 ϕ + 2 ,2) lim inf n | | F n α | | 1 5 ,3) lim inf n | | ϕ n α | | 1 5 .These results are the best possible.

Diophantine equations with linear recurrences An overview of some recent progress

Umberto Zannier (2005)

Journal de Théorie des Nombres de Bordeaux

We shall discuss some known problems concerning the arithmetic of linear recurrent sequences. After recalling briefly some longstanding questions and solutions concerning zeros, we shall focus on recent progress on the so-called “quotient problem” (resp. " d -th root problem"), which in short asks whether the integrality of the values of the quotient (resp. d -th root) of two (resp. one) linear recurrences implies that this quotient (resp. d -th root) is itself a recurrence. We shall also relate such...

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