Difference between revisions of "Rational approximation of famous numbers"

 
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==Introduction==
 
==Introduction==
The [[Rational approximation|Dirichlet's theorem]] shows that, for each irrational number <math>x\in\mathbb R</math>, the inequality <math>\left|x-\frac pq\right|<\frac 1{q^2}</math> has infinitely many solutions. On the other hand, sometimes it is useful to know that <math>x</math> cannot be approximated by rationals too well, or, more precisely, that <math>x</math> is not a [[Liouvillian number]].
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The [[Rational approximation|Dirichlet's theorem]] shows that, for each irrational number <math>x\in\mathbb R</math>, the inequality <math>\left|x-\frac pq\right|<\frac 1{q^2}</math> has infinitely many solutions. On the other hand, sometimes it is useful to know that <math>x</math> cannot be approximated by rationals too well, or, more precisely, that <math>x</math> is not a [[Liouvillian number]], i.e., that for some power <math>M<+\infty</math>, the inequality <math>\left|x-\frac pq\right|\ge \frac 1{q^M}</math> holds for all sufficiently large denominators <math>q</math>. So, how does one show that a number is not Liouvillian? The answer is given by the following
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==Main theorem==
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''
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Suppose that there exist <math>\beta>\mu>1</math>, <math>Q>1</math>
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and a sequence of rational numbers <math>\frac {P_n}{Q_n}</math> such that for all <math>n</math>, <math>Q_n\le Q^n</math> and
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<math>Q^{-\beta n}< \left|x-\frac {P_n}{Q_n}\right|<Q^{-\mu n}</math>. Then, for every <math>M>\frac\beta{\mu-1}</math>, the inequality <math>\left|x-\frac pq\right|<\frac 1{q^M}</math> has only finitely many solutions.
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''
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----
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The exact formulation of the main theorem in this article is fitted to the Beukers proof of the non-Liouvillian character of <math>\pi</math> but the general spirit of all such theorems is the same: roughly speaking, they tell you that in order to show that <math>x</math> cannot be approximated by rationals too well, one needs to find plenty of good but not too good rational approximations of <math>x</math>.
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==Proof of the Main Theorem==

Revision as of 11:23, 26 June 2006

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Introduction

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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