Dirichlet's approximation theorem: Difference between revisions

Revision as of 11:45, 26 September 2013

Revision as of 09:57, 5 May 2014

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:<math> \left | q \alpha -p \right | \le \frac{1}{N+1} </math>

This is a foundational result in [[diophantine approximation]], showing that any real number has a sequence of good rational approximations: in fact an immediate consequence is that for a given irrational α, the inequality

:<math> \left | \alpha -\frac{p}{q} \right | < \frac{1}{q^2} </math>

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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http://jeff560.tripod.com/p.html for a number of historical references.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>

Another simple proof of the Dirichlet's approximation theorem is based on [[Minkowski's_theorems theorem|Minkowski's Theorem]] applied to the set <math>S = \{ (x,y) \in \R^2; -N-\frac{1}{2} \leq x \leq N+\frac{1}{2}, \vert \alpha x - y \vert \leq \frac{1}{N} \} </math>. Since the volume of <math>S</math> is greater than <math>4</math>, [[Minkowski's_theorems theorem|Minkowski's Theorem]] establishes the existence of a non-trivial point with integral coordinates. This proof extends naturally to simultaneous approximations by considering the set: <math>S = \{ (x,y_1, \dots, y_d) \in \R^{1+d}; -N-\frac{1}{2} \le x \le N+\frac{1}{2}, |\alpha x - y_i| \le \frac{1}{N^{1/d}} \} </math>.

==See also==