Dirichlet's approximation theorem: Difference between revisions

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:<math> \left| q \alpha -p \right| < \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 equation

This theorem is a consequence of the [[pigeonhole principle]], and the one who popularized its use {{cn}}.

An immediate consequence is that for a given irrational α, the equation

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

is satisfied by infinitely many integers ''p'' and ''q''. This corollary shows that the [[Thue–Siegel–Roth theorem]], a result in the other direction, provides essentially the tightest possible bound, in the sense that approximation of [[algebraic number]]s cannot be improved by increasinglowering the exponent 2 + &epsilon; beyond 2.

==Method of proof==

This theorem is a consequence of the [[pigeonhole principle]]. 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http://jeff560.tripod.com/p.html for a number of historical references.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>

==References==

* [[Wolfgang M. Schmidt]]. ''Diophantine approximation''. Lecture Notes in Mathematics 785. Springer. (1980 [1996 with minor corrections])

* Wolfgang M. Schmidt.''Diophantine approximations and Diophantine equations'', Lecture Notes in Mathematics, Springer Verlag 2000

==Notes==

==External links==