## 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

### Keywords:

rational approximation; algebraic number

Zbl 0119.28103
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