Rational approximation of famous numbers

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The Dirichlet's theorem shows that, for each irrational number $x\in\mathbb R$, the inequality $\left|x-\frac pq\right|<\frac 1{q^2}$ has infinitely many solutions. On the other hand, sometimes it is useful to know that $x$ cannot be approximated by rationals too well, or, more precisely, that $x$ is not a Liouvillian number, i.e., that for some power $M<+\infty$, the inequality $\left|x-\frac pq\right|\ge \frac 1{q^M}$ holds for all sufficiently large denominators $q$. So, how does one show that a number is not Liouvillian? The answer is given by the following

Main theorem

Suppose that there exist $\beta>\mu>1$, $Q>1$ and a sequence of rational numbers $\frac {P_n}{Q_n}$ such that for all $n$, $Q_n\le Q^n$ and $Q^{-\beta n}< \left|x-\frac {P_n}{Q_n}\right|<Q^{-\mu n}$. Then, for every $M>\frac\beta{\mu-1}$, the inequality $\left|x-\frac pq\right|<\frac 1{q^M}$ has only finitely many solutions.

The exact formulation of the main theorem in this article is fitted to the Beukers proof of the non-Liouvillian character of $\pi$ but the general spirit of all such theorems is the same: roughly speaking, they tell you that in order to show that $x$ cannot be approximated by rationals too well, one needs to find plenty of good but not too good rational approximations of $x$.

Proof of the Main Theorem

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