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Simultaneous rational approximations to algebraic numbers. (English) Zbl 0098.26302

Let \(\beta_1, \beta_2, \dots,\beta_n\) be linearly independent numbers of a real algebraic number field of degree \(n+1\). By a study of the units of the field it is shown how to obtain all integral solutions of the set of inequalities: \[ q_0>0, \quad\gcd(q_0,q_1,\dots,q_n)=1,\quad | \beta_j/\beta_0-q_j/q_0| < Cq_0^{-1-1/n} \] for fixed \(C\). In particular, there are no solutions if the fixed number \(C\) is small enough. It is shown, further, that for an appropriate \(C\) there are infinitely many integer solutions of \[ | q_0\beta_j-q_j\beta_0| <C\,q_0^{-1/n}(\log q_0)^{-1/(n-1)}\quad (1\leq j\leq n-1),\quad | q_0\beta_n-q_n\beta_0| <C\,q_0^{-1/n}. \] This last result strengthens a theorem of J. W. S. Cassels and H. P. F. Swinnerton-Dyer [Philos. Trans. R. Soc. Lond., Ser. A 248, 73–96 (1955; Zbl 0065.27905)].
Reviewer: J. W. S. Cassels

MSC:

11J13 Simultaneous homogeneous approximation, linear forms
11J68 Approximation to algebraic numbers

Citations:

Zbl 0065.27905
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References:

[1] J. W. S. Cassels and H. P. F. Swinnerton-Dyer, On the product of three homogeneous linear forms and the indefinite ternary quadratic forms, Philos. Trans. Roy. Soc. London. Ser. A. 248 (1955), 73 – 96. · Zbl 0065.27905
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