Dirichlet's approximation theorem: Difference between revisions

Revision as of 22:10, 25 September 2022

Revision as of 22:11, 25 September 2022

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Another simple proof of the Dirichlet's approximation theorem is based on [[Minkowski's theorem]] applied to the set

: <math>S = \left\{ (x,y) \in \R^2; : -N-\frac{1}{2} \leq x \leq N+\frac{1}{2}, \vert \alpha x - y \vert \leq \frac{1}{N} \right\}. </math>

Since the volume of <math>S</math> is greater than <math>4</math>, [[Minkowski's theorem]] establishes the existence of a non-trivial point with integral coordinates. This proof extends naturally to simultaneous approximations by considering the set