Approximation diophantienne (théorie de Markoff)

Abstract : Towards 1880, A. Markoff gave precisions about structure of the set of approximation constants greater than 1/3 for irrational numbers. This theory establishes links between constants, arithmetical minima for quadratic forms, and the solutions of the diophantine equation x2 + y2 +z2 = 3xyz. The present dissertation generalizes the original formalism built by Markoff. It introduces the notion of (a, r,E)-theory of Markoff, among which the (2,0,-1) theory is the original Markoff theory. The corresponding diophantine equation is given with an interpretation for the whole calculus. From that is derives the resolution of the diophantine equation x2 + y2 +z2 = (a +1)xyz and some arborescent constructions. For the systematic research of holes in the Markoff's spectra, the author gives confirmation for the results of Schecker and Freiman, concerning the Hall's ray. He gives examples and gives confirmation for some results of Kinney and Pitcher
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Serge Perrine. Approximation diophantienne (théorie de Markoff). Mathématiques générales [math.GM]. Université Paul Verlaine - Metz, 1988. Français. ⟨NNT : 1988METZ026S⟩. ⟨tel-01775765⟩

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