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Approximation by Algebraic Numbers
Preface Frequently used notation 1. Approximation by rational numbers 2. Approximation to algebraic numbers 3. The classifications of Mahler and Koksma 4. Mahler's conjecture on S-numbers 5.…
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On Exponents of Homogeneous and Inhomogeneous Diophantine Approximation
In Diophantine Approximation, inhomogeneous problems are linked with homogeneous ones by means of the so-called Transference Theorems. We revisit this classical topic by introducing new exponents of…
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Exponents of Diophantine Approximation and Sturmian Continued Fractions
Soient ( un nombre reel et n un entier strictement positif. Nous definissons quatre exposants d'approximation diophantienne, qui viennent completer les exposants ω n (ξ) et ω* n (ξ) definis par…
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Bounds for the solutions of Thue-Mahler equations and norm form equations
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Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers
This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based…
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On the complexity of algebraic numbers I. Expansions in integer bases
Let b ≥ 2 be an integer. We prove that the b-ary expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are…
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On the number of solutions of the generalized Ramanujan-Nagell equation
Let D 1 and D 2 be coprime positive integers and let k be an odd positive integer coprime with D 1 D 2 . We consider the Diophantine equation D 1 x 2 + D 2 = k n in the unknowns x≥1, n≥1. We give a…
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On the complexity of algebraic numbers, II. Continued fractions
The continued fraction expansion of an irrational number $\alpha$ is eventually periodic if and only if $\alpha$ is a quadratic irrationality. However, very little is known regarding the size of the…
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Diophantine approximation and Cantor sets
We provide an explicit construction of elements of the middle third Cantor set with any prescribed irrationality exponent. This answers a question posed by Kurt Mahler.
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A NOTE ON INHOMOGENEOUS DIOPHANTINE APPROXIMATION
  • Y. Bugeaud
  • Mathematics
    Glasgow Mathematical Journal
  • 1 January 2003
Let $\alpha$ be an irrational number. We determine the Hausdorff dimension of sets of real numbers which are close to infinitely many elements of the sequence $(\{n\alpha\})_{n\,{\ge}\,1}$.
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