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

Keywords:

diophantine approximation

Zbl 0065.27905
Full Text:

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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