Dirichlet's approximation theorem: Difference between revisions

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In [[mathematics]], '''Dirichlet's theorem on [[diophantine approximation]]''', also called '''Dirichlet's approximation theorem''', states that for any [[real number]] αα, and any [[positive integer]] ''n'', there is some positive integer ''m'' ≤≤ ''n'' , such that the difference between ''m''αα 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 αα, at least one of the first five integer multiples of αα, namely

:1α, 2α, 3α, 4α, 5α,

:1α, 2α, 3α, 4α, 5α,

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

Dirichlet's approximation theorem shows that [[Roth's theorem]] is best possible in the sense that the occurring exponent cannot be increased, and thereby improved, to -2.