Diophantine Geometry: An Introduction
《Diophantine Geometry: An Introduction》 · 第 299–300 页 · Part D「Diophantine Approximation and Integral Points on Curves」，历史结果表
In general, the approximation exponent of a real number $\alpha\in\mathbb R$ is defined to be the smallest number $r(\alpha)$ with the property that for any exponent $e>r(\alpha)$, the inequality $|p/q-\alpha|\leq q^{-e}$ has only finitely many solutions in rational numbers $p/q\in\mathbb Q$. The exponent in Liouville's theorem for algebraic numbers of degree $d$ has been successively improved by a number of mathematicians, as indicated in the following brief table: Thue, 1909, $r(\alpha)\leq\frac12d+1$.