Roth's theorem in Diophantine approximation (1955) is a well known milestone. It has been generalised in the case of number fields for simultaneous approximations considering several places.
It is an ineffective result, in fact there is no known computable upper bound on the height of the good approximants. Altough there are several partial results (Bombieri-van der Poorten, Gross, Corvaja...) on the number of good approximants.
What are the known applications of Roths theorem in arithmetic geometry? (I am referring to the proved ineffective version, not to the potential applications of an effective version).
I am aware of the following consequences:
- Siegel theorem on integral points. Altough Corvaja and Zannier (2002) found a shorter proof by using the Schmidt subspace theorem.
- Ineffective finiteness for the solutions of the S-unit equation. This is a special case of the previous point and there are more recent techniques (Baker, Evertse...) that give explicit bounds.
- Ineffective finiteness for the solutions of Thue's equation.
Am I missing something in my list? In general Schmidt subspace theorem is considered much more useful of Roth's theorem in arithmetic geometry, but afterall the latter is a higher-dimensional generalisation of Roth's result.