Dirichlet's approximation theorem: Difference between revisions

Revision as of 22:10, 25 September 2022

Revision as of 22:10, 25 September 2022

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:<math> \left | q \alpha -p \right | \leq \frac{1}{\lfloor N\rfloor+1} < \frac{1}{N}. </math>

Here <math> [\lfloor N\rfloor </math> represents the [[integer part]] of <math> N </math>.

This is a fundamental 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