New applications of Diophantine approximations to Diophantine equations.
نویسندگان
چکیده
منابع مشابه
Diophantine Approximations, Diophantine Equations, Transcendence and Applications
This article centres around the contributions of the author and therefore, it is confined to topics where the author has worked. Between these topics there are connections and we explain them by a result of Liouville in 1844 that for an algebraic number α of degree n ≥ 2, there exists c > 0 depending only on α such that | α− p q |> c qn for all rational numbers p q with q > 0. This inequality i...
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The first course is devoted to the basic setup of Diophantine approximation: we start with rational approximation to a single real number. Firstly, positive results tell us that a real number x has “good” rational approximation p/q, where “good” is when one compares |x − p/q| and q. We discuss Dirichlet’s result in 1842 (see [6] Course N◦2 §2.1) and the Markoff–Lagrange spectrum ([6] Course N◦1...
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© Bulletin de la S. M. F., 1965, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http://smf. emath.fr/Publications/Bulletin/Presentation.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impress...
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For a real number x, let ||x|| denote the distance from x to the nearest integer. Suppose x1 < x2 < x3 are positive integers with gcd(x1, x2, x3) = 1. This paper proves the following: if (x1, x2, x3) 6= (1, 2, 3s) for an integer s and x3 6= x1+x2, or x3 = x1+x2 but x1 ≡ x2 (mod 3), then there is a real number t such that ||txi|| ≥ 1/3 (for i = 1, 2, 3). If (x1, x2, x3) = (1, 2, 3s) or x3 = x1 +...
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This brief survey deals with multi-dimensional Diophantine approximations in sense of linear form and with simultaneous Diophantine approximations. We discuss the phenomenon of degenerate dimension of linear sub-spaces generated by the best Diophantine approximations. Originally most of these results have been established by the author in [14, 15, 16, 17, 18]. Here we collect all of them togeth...
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ژورنال
عنوان ژورنال: MATHEMATICA SCANDINAVICA
سال: 1976
ISSN: 1903-1807,0025-5521
DOI: 10.7146/math.scand.a-11641