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RATIONAL APPROXIMATIONS TO ALGEBRAIC NUMBERS
Mathematics of the USSR-Izvestiya, 1971In this article we derive a new effective estimate of rational approximations to algebraic numbers simultaneously in an Archimedian and several non-Archimedian metrics.
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Approximation of Algebraic Numbers
2008The following notes are an enriched exposition of the material which I presented during the C.I.M.E. summer school in Cetrato. My main goal is to illustrate the ideas behind the proofs of recent results generalizing the Sub-space Theorem in diophantine approximation. I have tried to keep a balance between avoiding certain technical details (which would
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Rational and algebraic approximations of algebraic numbers and their application
Science in China Series A: Mathematics, 1997Mit \(D,x_0,y_0\in\mathbb N\) werde \(u:= x_0+y_0\sqrt{-D}\), \(\overline u:= x_0-y_0\sqrt{-D}\), \(\varepsilon:= | u| +x_0\), \(\overline\varepsilon:= | u| -x_0\), \(w:= \overline u/u\) gesetzt und \(\varphi\in ] -\pi,0[\) bezeichne den Hauptwert von arg\( w\).
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Simultaneous approximation of algebraic numbers
Mathematical Notes, 1992Let 1, \(\theta_ 1, \dots, \theta_ s\) \((s \geq 2)\) be a basis of a purely algebraic field \(\mathbb{K}\) of degree \(s+1\). The author proves the following theorem: assume that a natural number simultaneously approximates the number \(\theta_ 1, \dots, \theta_ s\), \[ \| q \theta_ i \| = \min_{a \in \mathbb{Z}} | q \theta_ i - a |0\) is some ...
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Mathematics of the USSR-Izvestiya, 1977
A new estimate of the solutions of the generalized Thue-Mahler equation is derived, which explicitly exhibits the influence of all fundamental parameters of the equation on the magnitude of the solutions. Also, an effective power sharpening is given of "Liouville's inequality" relating to the approximation of algebraic numbers by algebraic numbers of a
Kotov, S. V., Sprindzuk, V. G.
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A new estimate of the solutions of the generalized Thue-Mahler equation is derived, which explicitly exhibits the influence of all fundamental parameters of the equation on the magnitude of the solutions. Also, an effective power sharpening is given of "Liouville's inequality" relating to the approximation of algebraic numbers by algebraic numbers of a
Kotov, S. V., Sprindzuk, V. G.
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On the approximation of logarithms of algebraic numbers
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 1953Abstract A new identity is given by means of which infinitely many algebraic functions approximating the logarithmic function In x are obtained. On substituting numerical algebraic values for the variable, a lower bound for the distance of its logarithm from variable algebraic numbers is found.
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Simplifying method for algebraic approximation of certain algebraic numbers
Mathematical Notes, 2012Motivated by the famous Ljunggren's (written erroneously Ljungerren in the Introduction of the paper) Diophantine equation \(x^2+1=2y^4\), the solution of which is straightforwardly implied by the solution of the Thue equation \(x^4-4x^3y-6x^2y^2+4xy^3+y^4=\pm 1\), the authors study the more general Thue equation \[ f(x,y)=tx^4-4sx^3y-6tx^2y^2+4sxy^3 ...
Xia, Jingbo, Chen, Jianhua, Zhang, Silan
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On the approximation of quadratic algebraic numbers
Studia Scientiarum Mathematicarum Hungarica, 2005In the paper we construct such second order linear recursive sequences G and H of rational integers that with their terms |a -Gn+1 /H n| < 1/ (\sqrtvDH2n) holds for every positive integer n, where a denotes a real quadratic algebraic integer of discriminant D.
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Rational approximations to algebraic numbers
Mathematika, 1957It was proved by Roth in a recent paper that if α is any real algebraic number, and if K > 2, then the inequalityhas only a finite number of solutions in relatively prime integers p, q (q > 0) The object of the present paper is to prove that the lower bound for κ can be reduced if conditions are imposed on p and q. The result obtained is as follows.
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On a simultaneous approximation of logarithms and algebraic powers of algebraic numbers
Mathematical Notes, 1994Let \(\theta\in \mathbb{C}\) be an arbitrary transcendental number. Denote \(\mathbb{Q}_1= \mathbb{Q}(\theta)\) and \(\mathbb{J}_1= \mathbb{Z}[\theta]\). Let \(\mathbb{Q}^*_1\) be an algebraic extension of \(\mathbb{Q}_1\) of finite degree generated by the numbers \(\theta\) and \(\omega_1\), where \(\omega_1\) is a root of an irreducible polynomial in
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