# Thue-Siegel-Roth theorem

If $\alpha$ is an irrational algebraic number and $\delta>0$ is arbitrarily small, then there are only finitely many integer solutions $p$ and $q>0$ ($p$ and $q$ being co-prime) of the inequality
$$\left|\alpha-\frac pq\right|<\frac{1}{q^{2+\delta}}.$$
This theorem is best possible of its kind; the number 2 in the exponent cannot be decreased. The Thue–Siegel–Roth theorem is a strengthening of the Liouville theorem (see Liouville number). Liouville's result has been successively strengthened by A. Thue , C.L. Siegel  and, finally, K.F. Roth . Thue proved that if $\alpha$ is an algebraic number of degree $n\geq3$, then the inequality
$$\left|\alpha-\frac pq\right|<\frac{1}{q^\nu}$$
has only finitely many integer solutions $p$ and $q>0$ ($p$ and $q$ being co-prime) when $\nu>(n/2)+1$. Siegel established that Thue's theorem is true for $\nu>2n^{1/2}$. The final version of the theorem stated above was obtained by Roth. There is a $p$-adic analogue of the Thue–Siegel–Roth theorem. The results listed above are proved by non-effective methods (see Diophantine approximation, problems of effective).