Dirichlet's approximation theorem: Difference between revisions

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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http://jeff560.tripod.com/p.html for a number of historical references.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 outline''': Let <math>\alpha</math> be an irrational number and <math>n</math> be an integer. For every <math>k=0, 1, ..., nN</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>nN</math> smaller intervals of measure <math>\frac{1}{nN}</math>. Now, we have <math>nN+1</math> numbers <math>x_0,x_1,...,x_nx_N</math> and <math>nN</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:

: <math>|(j-i)\alpha-(m_j-m_i)|=|j\alpha-m_j-(i\alpha-m_i)|=|x_j-x_i|< \frac{1}{nN}</math>

Dividing both sides by <math>j-i</math> will result in:

: <math>\left|\alpha-\frac{m_j-m_i}{j-i}\right|< \frac{1}{(j-i)nN}\le \frac{1}{\left(j-i\right)^2}</math>

And we proved the theorem.