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.