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'' < ''n'' , such that the difference between ''m''&alpha; and the nearest integer is at most 1/(''n'' multiple+ 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.

:''m''&alpha; or &minus;''m''&alpha;

has relatively small [[fractional part]] (in other words, the multiples of &alpha; can't stay too far away from integers). In quantitative terms, the difference of one of the first ''N'' multiples and some integer must take a value at most

:1/(''N'' + 1).

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