# Well known applications of Roth's theorem

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:

1. Siegel theorem on integral points. Altough Corvaja and Zannier (2002) found a shorter proof by using the Schmidt subspace theorem.
2. 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.
3. 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.

If nothing else we can use Roth's theorem to generalize Liouville's construction of transcendental numbers.

Liouville noted that numbers such as $$\lambda = \sum_{k=1}^\infty 1 / 10^{k!}$$ are irrational because they have rational approximations $$p_n/q_n$$ with $$0 < \left| \lambda - (p_n/q_n) \right| \sim 1 / q_n^n$$, while an irrational algebraic number $$x$$ of degree $$N>1$$ satisfies $$\left| x - (p/q) \right| \gg q^{-N}$$. More generally this works for $$\lambda = \sum_{a=1}^\infty a_k / b_k^{e_k}$$ for some bounded positive integers $$a_k$$ and exponents $$e_k$$ with $$\limsup_k e_{k+1} / e_k = +\infty$$.

Thanks to Roth, this last condition can be relaxed to $$\limsup_k e_{k+1} / e_k = \theta$$ for some $$\theta > 2$$. So for example the number $${\small \rho = 3.1{\tiny0}4{\tiny00000}1{\tiny00000000000000000}5{\tiny00000000000000000000000000000000000000000000000000000}9{\tiny00000}\ldots }$$ (with the digits $$d_k$$ of $$\pi = 3 + \sum_{k=1}^\infty d_k/10^k$$ spread out to $$\rho = 3 + \sum_{k=1}^\infty d_k/10^{3^{k-1}}$$) is transcendental; here $$\theta = 3$$.

As far as I know and as you have already pointed out above, Roth's theorem has been later extended in higher dimensions (as very often happens in mathematics) by Schmidt's subspace theorem and also by considering the p-adics, since 1958 Ridout's paper "The p-adic generalization of the Thue-Siegel-Roth theorem" (see https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0025579300001339).
Now, the main (intrinsic, maybe) limit of Roth's theorem has been presented by Hindry&Silverman (2000) in "Diophantine Geometry: An Introduction", pp. 299-366, so basically we cannot efficiently bound the key parameters of Lang's conjecture $$\lvert \alpha - \frac{p}{q} \rvert <\frac{1}{q^2 \dot log(q)^{1+\epsilon}}$$ (i.e., $$p,q \in \mathbb{Z}$$ so that $$\frac{p}{q} \in \mathbb{Q}$$ by construction) when $$\alpha \in \mathbb{R}$$ (including transcendental numbers as well). Furthermore, the same reference is useful to underline that Roth's theorem is a key tool in proving the Mordell conjecture (i.e., Faltings's theorem).
On the other hand, I have just seen a recent preprint on arxiv which is focused on the (possible) implications of R.T. to "a weakened non-effective version of the ABC Conjecture in certain cases relating to roots", basing on the mentioned, remakable, Bombieri-van der Poorten's results, but I have not still given at it a proper look. I am referring to https://arxiv.org/abs/2208.14354.\ Moreover, Faltings&Ching-Li's "Arithmetic Geometry" concerns the application of Roth's theorem in the study of heights of algebraic points (a measurement of their complexity), so this could even be another link from R.T. to the complexity of algebraic points $$\rightarrow$$ algebraic geometry.