Dirichlet approximation theorem

This article deals with diophantine approximation. For other uses of this term, see Dirichlet's Theorem. Dirichlet's approximation theorem or Dirichlet's theorem on D phantic approximation asserts that for any real number α and any positive integer N, there exist integers p and q such that 1 ≤ q ≤ N and | q α & # x2212; p | < 1 N {\displaystyle \left|q\alpha -p\right|<{\frac {1}{N}}}

This is a fundamental result of a diophantine approach, showing that any real number has a succession of good rational approximations: in fact, an immediate consequence is that given an irrational number α, the inequality | α & # x2212; p q | < 1 q 2 {\displaystyle \left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{2}}}}

is satisfied for infinite integers p and q.

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