In mathematics, Dirichlet's theorem on diophantine approximation (Dirichlet's approximation theorem) states that for any real number α, some integer multiple
- mα or −mα
has relatively small fractional part (in other words, the multiples of α can't stay too far away from integers). In quantitative terms, the difference of one of the first N multiples and some integer must take a value at most
- 1/(N + 1).
This is a consequence of the pigeonhole principle.