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On rational approximations of algebraic numbers \(\root3\of D\). (Russian) Zbl 1047.11068

In [Proc. Int. Congr. Math. 1958, 203–210 (1960; Zbl 0119.28103)], K. Roth proved the following theorem: Let \(\alpha\) be an algebraic number, \(\operatorname{deg}\alpha=n\geq 3\), and let \(\delta\) be a positive number. Then the inequality \(| \alpha-\frac{p}{q}| <\frac{1}{q^{2+\delta}}\) has only finitely many solutions in \(p\in \mathbb Z\), \(q\in\mathbb N\).
The author considers the case in which \(\alpha=\root3\of D\), \(D\in\mathbb N\). He gives a new proof of Roth’s theorem for this case and proves that the inequality \(| \alpha-\frac{p}{q}| < \frac{1}{q^{2}\log^{1+\delta}q}\) (\(0<\delta<1\)) has only finitely many solutions in \(p\in\mathbb Z\), \(q\in\mathbb N\).

MSC:

11J68 Approximation to algebraic numbers

Citations:

Zbl 0119.28103
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