##  [Roth's Theorem](/roths-theorem) 

  ##  [Roth's Theorem](https://puremath.quantumdictionary.io/roths-theorem-0) 

  

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- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

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.