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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.
==Simultaneous Version==
The simultaneous version of the Dirichlet's approximation theorem states that given real numbers <math>\alpha_1, ..., \alpha_d</math> and a natural number <math>N</math> then there are integers <math>p_1, ..., p_d, q \in \mathbb{Z}</math> such that <math>\left | \alpha - \frac{p_i}{q}
\right | \leq \frac{1}{p N^{1/d}} </math>
==Method of proof==
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