Dirichlet's approximation theorem: Difference between revisions

Revision as of 22:38, 20 August 2017

Revision as of 00:35, 2 July 2018

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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 bound on rational approximation of [[algebraic number]]s cannot be improved by increasing the exponent beyond 2.

==Simultaneous Versionversion==

The simultaneous version of the Dirichlet's approximation theorem states that given real numbers <math>\alpha_1, ...\ldots, \alpha_d</math> and a natural number <math>N</math> then there are integers <math>p_1, ...\ldots, p_d, q\in\Z,1\le q\leq N</math> such that <math>\left|\alpha_i-\frac{p_i}q \right| \le \frac1{qN^{1/d}}.</math>

==Method of proof==