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In [[number theory]], '''[[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 exists integers ''p'' and ''q'' such that 1 ≤ ''q'' ≤ ''N'' and
:<math> \left | q \alpha -p \right | \le \frac{1}{N+1} </math>
This is a foundational result in [[diophantine approximation]], showing that any real number has a sequence of good rational approximations: in fact an immediate consequence is that for a given irrational α, the inequality
:<math> \left | \alpha -\frac{p}{q} \right | < \frac{1}{q^2} </math>
is satisfied by infinitely many integers ''p'' and ''q''. This corollary also shows that the [[Thue–Siegel–Roth theorem]], a result in the other direction, provides essentially the tightest possible bound, in the sense that the limits on rational approximation of [[algebraic number]]s cannot be improved by lowering the exponent 2 + ε beyond 2.
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