Dirichlet's approximation theorem: Difference between revisions

Revision as of 00:08, 2 February 2025

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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

:<math> 0<\left | \alpha -\frac{p}{q} \right | < \frac{1}{q^2} </math>

is satisfied by infinitely many integers ''p'' and ''q''. This shows that any irrational number has [[irrationality measure]] at least 2.