Dirichlet's approximation theorem: Difference between revisions

Revision as of 19:53, 6 November 2005

Revision as of 02:19, 9 May 2006

Line 1:

In [[mathematics]], '''Dirichlet's theorem''' on [[diophantine approximation]]''', also called ('''Dirichlet's approximation theorem'''), states that for any [[real number]], α, and any [[positive integer]], ''n'', there is some positive integer, ''m'' ≤ ''n'' , such that the difference between ''m''α and the nearest integer is at most 1/(''n'' + 1). This is a consequence of the [[pigeonhole principle]].

For example, no matter what value is chosen for α, at least one of the first five integer multiples of α, namely

Line 7:

will be within 1/6 of an integer, either above or below. Likewise, at least one of the first 20 integer multiples of α will be within 1/21 of an integer.

Dirichlet's approximation theorem shows that [[Roth's theorem]] is best possible.

This is a consequence of the [[pigeonhole principle]].

==External link==