Editing Roth's theorem (section)

== Statement ==

Roth's theorem states that every [[Irrational number|irrational]] algebraic number <math>\alpha</math> has [[approximation exponent]] equal to 2. This means that, for every <math>\varepsilon>0</math>, the [[inequality (mathematics)|inequality]]

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

can have only [[finite set|finitely many]] solutions in [[coprime]] [[integer]]s <math>p</math> and <math>q</math>. Roth's [[mathematical proof|proof]] of this fact resolved a [[conjecture]] by Siegel. It follows that every irrational algebraic number α satisfies

:<math>\left|\alpha - \frac{p}{q}\right| > \frac{C(\alpha,\varepsilon)}{q^{2 + \varepsilon}}</math>

with <math>C(\alpha,\varepsilon)</math> a positive number depending only on <math>\varepsilon>0</math> and <math>\alpha</math>.