Roth's Theorem

A fundamental result in Diophantine approximation stating that any irrational algebraic number α cannot be approximated by rationals p/q arbitrarily closely: for every ε>0, the inequality |α−p/q|<1/q^{2+ε} has only finitely many rational solutions p/q. Equivalently, the approximation exponent of an algebraic irrational is 2.