Dirichlet’s approximation theorem – Wikipedia

irrational α, the inequality | α − p q | < 1 q 2 {\displaystyle \left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{2}}}} \left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{2}}}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 numbers cannot be improved by increasing the exponent beyond 2.

Source: Dirichlet’s approximation theorem – Wikipedia

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