Simultaneous rational approximation to the successive powers of a real number
نویسندگان
چکیده
منابع مشابه
SIMULTANEOUS RATIONAL APPROXIMATION TO THE SUCCESSIVE POWERS OF A REAL NUMBER by Michel LAURENT
Let n be an integer ≥1 and let θ be a real number which is not an algebraic number of degree ≤ n/2 . We show that there exist >0 and arbitrary large real numbers X such that the system of linear inequalities |x0|≤X and |x0θ−xj |≤ X−1/ n/2 for 1≤j≤n, has only the zero solution in rational integers x0,...,xn. This result refines a similar statement due to H. Davenport and W. M. Schmidt, where the...
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We apply Padé approximation techniques to deduce lower bounds for simultaneous rational approximation to one or more algebraic numbers. In particular, we strengthen work of Osgood, Fel’dman and Rickert, proving, for example, that max {∣∣∣√2− p1/q∣∣∣ , ∣∣∣√3− p2/q∣∣∣} > q−1.79155 for q > q0 (where the latter is an effective constant). Some of the Diophantine consequences of such bounds will be d...
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ژورنال
عنوان ژورنال: Indagationes Mathematicae
سال: 2003
ISSN: 0019-3577
DOI: 10.1016/s0019-3577(03)90070-x