Liouville approximation theorem

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

\\left|\\alpha-\\frac{p}{q}\\right|>\\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 $p/q$ , the inequality
$\\left|\\alpha-\\frac{p}{q}\\right|>c_{0}q^{-1-\\epsilon-n/2}$
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 $p/q$ , the inequality
$\\left|\\alpha-\\frac{p}{q}\\right|>c_{1}q^{-\\lambda},\\qquad\\lambda={\\min}_{t=1,%\\ldots,n}\\left(\\frac{n}{t+1}+t\\right)+\\epsilon$
holds.
•
Dyson: If $\\alpha$ is an algebraic number of degree $n>3$ , then there is a constant $c_{2}=c_{2}(\\alpha,\\epsilon)>0$ such that for all rational numbers $p/q$ with $q>c_{2}$ , the inequality
$\\left|\\alpha-\\frac{p}{q}\\right|>q^{-\\sqrt{2n}-\\epsilon}$
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 $p/q$ , the inequality
$\\left|\\alpha-\\frac{p}{q}\\right|>c_{3}q^{-2-\\epsilon}$
holds.