Liouville approximation theoremLiouvillesTheorem2001-10-15 20:20:432003-02-01 12:53:59Theoremnumber theory\usepackage{amssymb}
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\usepackage{xypic}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:
\begin{itemize}
\item 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.
\item 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.
\item 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.
\item 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.
\end{itemize}