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 mα and the nearest integer is at most 1/(n + 1). This is a consequence of the pigeonhole principle. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers. ... In mathematics, the real numbers may be described informally in several different ways. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... The inspiration for the name of the principle: pigeons in holes. ...

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

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 α 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. There are two major results of Klaus Roth in mathematics which go by the name of Roths theorem: The Thue-Siegel-Roth theorem in Diophantine approximation, which concerns the rarity to which an irrational algebraic number can be approximated by a rational number; and Roths theorem in arithmetic...

External link

• PlanetMath page