# Liouville approximation 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   $p/q,(p,q)=1$, 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.

Title Liouville approximation theorem  LiouvilleApproximationTheorem 2013-03-22 11:45:45 2013-03-22 11:45:45 KimJ (5) KimJ (5) 13 KimJ (5) Theorem msc 11J68 msc 46-01 msc 46N40 ExampleOfTranscendentalNumber