Dirichlet's approximation theorem: Difference between revisions

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==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}, q \leq N</math> such that <math>\left | \alpha - \frac{p_i}{q}

\right | \leq \frac{1}{p N^{1/d}} </math>

==Method of proof==