# Thue-Siegel-Roth theorem

2010 Mathematics Subject Classification: Primary: 11J68 [MSN][ZBL]

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).

How to Cite This Entry:
Thue–Siegel–Roth theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Thue%E2%80%93Siegel%E2%80%93Roth_theorem&oldid=23086