I saw that there are many "applications" questions in Mathoverflow; so hopefully this is an appropriate question. I was rather surprised that there were only five questions at Mathoverflow so far with the tag diophantine-approximation, while there are almost 900 questions on number theory overall. It is my intention to promote the important subject a little bit by asking one more question.
Question:
What are some striking applications of Baker's theorem on lower bounds for linear forms on logarithms of algebraic numbers?
If, for example, I were in a discussion with a person who has no experience with diophantic approximation, to impress upon the person the importance of Baker's theorem I would cite the following two examples:
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Giving effective bounds for solutions of (most of the time exponential) diophantine equations under favorable condition. For example, Tijdeman's work on the Catalan conjecture, or giving effective bounds for Siegel's theorem, Fermat's last theorem, Falting's theorem, etc., in certain cases.
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Transcendence results which are significant improvements over Gelfond-Schneider. In particular, the theorem that if $\alpha_1, \ldots, \alpha_n$ are $\mathbb{Q}$-linearly independent, then their exponentials are algebraically independent over $\mathbb Q$. I would cite the expose of Waldschmidt for more details.
These are, to me, quite compelling reasons to study Baker's theorem. But as I do not know much more on the subject, I would run out of arguments after these two. I would appreciate any more striking examples of the power of Bakers' theorem. This is 1. for my own enlightenment, 2., for future use if such an argument as I hypothesized above actually happens, 3. To promote the subject of diophantine approximation in this forum, especially in the form of Baker's theorem.
Best Answer
A quantitative example of applications of linear forms in logarithms is the following result from [S.D. Adhikari, N. Saradha, T.N. Shorey, and R.Tijdeman, Transcendental infinite sums, Indag. Math. (N.S.) 12:1 (2001) 1--14; Theorem 4 and Corollary 4.1] which is cited in many other articles.
Theorem. Let $P(x)$ and $Q(x)$ be two polynomials with algebraic coefficients such that $Q(x)$ has simple rational zeros and no others. Let $\alpha$ be an algebraic number. Then, assuming the convergence of the series $$ S=\sum_{n=1}^\infty\frac{P(n)}{Q(n)}\alpha^n, $$ the number $S$ defined by it is either rational or transcendental. Furthermore, if all zeros of $Q(x)$ lie in $-1\le x<0$, then either $S=0$ or $S$ is transcendental.
The theorem gives an elegant criterion for deciding whether a number of this particular form is transcendental or not (like continued fractions allow us to decide whether a given number is from a quadratic field or not). However it is not applicable for the sums like $$ \zeta(3)=\sum_{n=1}\frac1{n^3}, $$ as factorisation of the denominator polynomial, $Q(x)=x^3$, involves multiple rational zeros.