Thue–Siegel–Roth theorem

Remarks

The proof builds an auxiliary polynomial in many variables with integer coefficients that vanishes to high order at $(\alpha, \dots, \alpha)$. Suppose there were many very good approximations with fast growing denominators. Then this polynomial would have to vanish to high order at a rational point too, and a nonvanishing lemma (Roth's lemma) rules this out. The proof is not effective: it gives a bound for the number of exceptional fractions, but not for their size. The exponent $2$ cannot be lowered, because Dirichlet's approximation theorem gives infinitely many $p/q$ with $\abs{\alpha - p/q} < 1/q^2$. The hypothesis that $\alpha$ is algebraic is necessary: a Liouville number has, for each exponent $\mu$, infinitely many approximations with error less than $1/q^{\mu}$. For example, $\alpha = \sqrt[3]{2}$ has only finitely many approximations with error less than $1/q^{2.01}$. The theorem sharpens Liouville's theorem on Diophantine approximation, where the exponent is the degree of $\alpha$, and the later results of Thue and Siegel.

Going deeper

For a student

Roth's theorem implies that a Thue equation $F(x, y) = m$, with $F$ an irreducible binary form of degree at least $3$, has only finitely many integer solutions. Its higher-dimensional generalization is the Schmidt subspace theorem. The theorem says that the irrationality exponent of each algebraic irrational is $2$, the same as for almost all real numbers.