In mathematics, Dirichlet's theorem on diophantine approximation (Dirichlet's approximation theorem) states that for any real number, α, and 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 5 integer multiples of α - 1α, 2α, 3α, 4α, 5α - will be within 1/6 of an integer.
This is a consequence of the pigeonhole principle.