Some remarks on Diophantine approximation by the Jacobi-Perron algorithm
Fritz Schweiger (2008)
Acta Arithmetica
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Fritz Schweiger (2008)
Acta Arithmetica
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Eugène Dubois, Ahmed Farhane, Roger Paysant-Le Roux (2004)
Acta Arithmetica
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Kentaro Nakaishi (2006)
Acta Arithmetica
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Richard B. Lakein (1973)
Monatshefte für Mathematik
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Schweiger, Fritz (2010)
Integers
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David J. Grabiner (1992)
Monatshefte für Mathematik
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M.S. Waterman, W.A. Beyer (1972)
Numerische Mathematik
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Yong-Gao Chen, Christian Elsholtz, Li-Li Jiang (2012)
Acta Arithmetica
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Avraham Bourla (2014)
Acta Arithmetica
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We establish arithmetical properties and provide essential bounds for bi-sequences of approximation coefficients associated with the natural extension of maps, leading to continued fraction-like expansions. These maps are realized as the fractional part of Möbius transformations which carry the end points of the unit interval to zero and infinity, extending the classical regular and backwards continued fraction expansions.
Jingcheng Tong (1991)
Monatshefte für Mathematik
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Leon Bernstein (1964)
Journal für die reine und angewandte Mathematik
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J.L. Davidson, J.O. Shallit (1991)
Monatshefte für Mathematik
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Yasushige Watase (2015)
Formalized Mathematics
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In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution (x, y) of the inequality |xθ − y| ≤ 1/x, where 0 is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. Finally, we formalize Dirichlet’s proof (1842) of existence of the solution [12], [1]. ...