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Thue-Siegel-Roth theorem : ウィキペディア英語版
Thue–Siegel–Roth theorem

In mathematics, the Thue–Siegel–Roth theorem, also known simply as Roth's theorem, is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that a given algebraic number \alpha may not have too many rational number approximations, that are 'very good'. Over half a century, the meaning of ''very good'' here was refined by a number of mathematicians, starting with Joseph Liouville in 1844 and continuing with work of , , , and .
== Statement ==

The Thue–Siegel–Roth theorem states that any irrational algebraic number \alpha has approximation exponent equal to 2, ''i.e.'', for given \epsilon>0, the inequality
:\left|\alpha - \frac\right| < \frac\right| > \frac{q^{2 + \epsilon}}
with C(\alpha,\epsilon) a positive number depending only on \epsilon>0 and \alpha.

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