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]]. | + | 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 |
| + | ==Main theorem== | ||
| + | '' | ||
| + | Suppose that there exist <math>\beta>\mu>1</math>, <math>Q>1</math> | ||
| + | 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 | ||
| + | <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. | ||
| + | '' | ||
| + | ---- | ||
| + | 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>. | ||
| + | ==Proof of the Main Theorem== | ||
Revision as of 10:23, 26 June 2006
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Introduction
The Dirichlet's theorem shows that, for each irrational number 
, the inequality 
has infinitely many solutions. On the other hand, sometimes it is useful to know that 
cannot be approximated by rationals too well, or, more precisely, that is not a Liouvillian number, i.e., that for some power 
, the inequality 
holds for all sufficiently large denominators 
. So, how does one show that a number is not Liouvillian? The answer is given by the following
Main theorem
Suppose that there exist 
, 
and a sequence of rational numbers 
such that for all 
, 
and
. Then, for every 
, the inequality 
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 
but the general spirit of all such theorems is the same: roughly speaking, they tell you that in order to show that cannot be approximated by rationals too well, one needs to find plenty of good but not too good rational approximations of .
