Dirichlet's approximation theorem
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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.
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.
