PlanetMath: Liouville approximation theorem

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Liouville approximation theorem

(Theorem)

Given $\alpha$ , a real algebraic number of degree $n \neq 1$ , there is a constant $c = c( \alpha ) > 0$ such that for all rational numbers , the inequality

$\displaystyle \left\vert \alpha - \frac{p}{q} \right\vert > \frac{c(\alpha )}{q^n}$

holds.

Many mathematicians have worked at strengthening this theorem:

Thue: If $\alpha$ is an algebraic number of degree $n \geq 3$ , then there is a constant $c_0 = c_0( \alpha , \epsilon ) > 0$ such that for all rational numbers , the inequality
$\displaystyle \left\vert \alpha - \frac{p}{q} \right\vert > c_0 q^{-1- \epsilon - n/2}$
holds.
Siegel: If $\alpha$ is an algebraic number of degree $n \geq 2$ , then there is a constant $c_1 = c_1( \alpha , \epsilon ) > 0$ such that for all rational numbers , the inequality
$\displaystyle \left\vert \alpha - \frac{p}{q} \right\vert > c_1 q^{- \lambda}, ... ...d \lambda = {\min}_{t=1,\ldots ,n} \left( \frac{n}{t+1} + t \right) + \epsilon$
holds.
Dyson: If $\alpha$ is an algebraic number of degree , then there is a constant $c_2 = c_2( \alpha , \epsilon ) > 0$ such that for all rational numbers with , the inequality
$\displaystyle \left\vert \alpha - \frac{p}{q} \right\vert > q^{- \sqrt{2n}- \epsilon }$
holds.
Roth: If $\alpha$ is an irrational algebraic number and $\epsilon > 0$ , then there is a constant $c_3 = c_3( \alpha , \epsilon ) > 0$ such that for all rational numbers , the inequality
$\displaystyle \left\vert \alpha - \frac{p}{q} \right\vert > c_3 q^{-2 - \epsilon }$
holds.