Dirichlet's approximation theorem: Difference between revisions

Revision as of 20:57, 11 September 2021

Revision as of 08:29, 23 January 2022

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

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. The Thue–Siegel–Roth theorem uses advanced techniques of number theory, but many simpler numbers such as the [[golden ratio]] <math>\frac{(1+\sqrt{5}}{)/2}</math> can be much more easily verified to be inapproximable beyond exponent 2. This exponent is referred to as the [[irrationality measure]].

==Simultaneous version==