In mathematics, Dirichlet's theorem on diophantine approximation (Dirichlet's approximation theorem) states that for any real number, α, and positive integer, n, there is some integer, m ≤ n , such that the difference between mα and the nearest integer is at most 1/(n + 1).
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.
This is a consequence of the pigeonhole principle.