Dirichlet's approximation theorem: Difference between revisions

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In [[mathematics]], '''Dirichlet's theorem''' on [[diophantine approximation]] ('''Dirichlet's approximation theorem''') states that for any [[real number]], &alpha;, and [[integer]], ''n'', there is some integer, ''m'' <&le; ''n'' , such that the difference between ''m''&alpha; and the nearest integer is at most 1/(''n'' + 1).

For example, no matter what value is chosen for &alpha;, at least one of the first 5 integer multiples of &alpha; - 1&alpha;, 2&alpha;, 3&alpha;, 4&alpha;, 5&alpha; - will be within 1/6 of an integer, either above or below. Likewise, at least one of the first 20 integer multiples of &alpha; will be within 1/21 of an integer.

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