Dirichlet's approximation theorem: Difference between revisions

Revision as of 03:40, 16 November 2022

Revision as of 23:45, 26 June 2023

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==Method of proof==

===Proof Byby Thethe Pigeonholepigeonhole Principleprinciple===

This theorem is a consequence of the [[pigeonhole principle]]. [[Peter Gustav Lejeune Dirichlet]] who proved the result used the same principle in other contexts (for example, the [[Pell equation]]) and by naming the principle (in German) popularized its use, though its status in textbook terms comes later.<ref>http://jeff560.tripod.com/p.html for a number of historical references.</ref> The method extends to simultaneous approximation.<ref>{{Springer|id=d/d032940|title=Dirichlet theorem}}</ref>

'''Proof Outlineoutline''': Let <math>\alpha</math> be an irrational number and <math>n</math> be an integer. For every <math>k=0, 1, ..., n</math> we can write <math>k\alpha=m_k + x_k</math> such that <math>m_k</math> is an integer and <math>0\le x_k <1</math>.

One can divide the interval <math>[0, 1)</math> into <math>n</math> smaller intervals of measure <math>\frac{1}{n}</math>. Now, we have <math>n+1</math> numbers <math>x_0,x_1,...,x_n</math> and <math>n</math> intervals. Therefore, by the pigeonhole principle, at least two of them are in the same interval. We can call those <math>x_i,x_j</math> such that <math>i < j</math>. Now:

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And we proved the theorem.

===Proof Byby Minkowski's theorem===

Another simple proof of the Dirichlet's approximation theorem is based on [[Minkowski's theorem]] applied to the set