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Simultaneous approximation of algebraic irrationalities

Authors:	B F Skubenko

Abstract:	This paper proves three theorems concerning the simultaneous approximation of numbers from a totally real algebraic number field. It is shown that for two given numbers ₁ and ₂ from a totally real algebraic number field, the constant ₁₂ can be explicitly calculated, this being the upper limit of the numbers c₁₂ such that the inequality max (q₁, q₂)(qc₁₂)^–1/2 holds for infinitely many natural numbers q; likewise for the constant a₁₂ such that the inequality q₁·q₂< a₁₂(q^logq) holds for infinitely many natural numbers q. It is shown that there exist n –1 numbers ₁, ..., _n–1 in an algebraic number field of degree n and discriminant d such that the inequality $\max \left( {\left\\| {q\theta _1 } \right\\|, ..., \left\\| {q\theta _{n - 1} } \right\\|} \right)< \left( {\gamma q} \right)^{ - \frac{1}{{n - 1}}}$ holds only for finitely many natural numbers q if $\gamma > 2^{ - \left {\tfrac{{n - 1}}{2}} \right]} \sqrt d$ . is fixed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 142–154, 1982.

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