Editing Roth's theorem (section)

== Proof technique ==

The proof technique involves constructing an [[auxiliary function|auxiliary]] multivariate [[polynomial]] in an arbitrarily large number of variables depending upon <math>\varepsilon</math>, leading to a [[proof by contradiction|contradiction]] in the presence of too many good approximations. More specifically, one finds a certain number of rational approximations to the irrational algebraic number in question, and then applies the function over each of these simultaneously (i.e. each of these rational numbers serve as the input to a unique variable in the expression defining our function). By its nature, it was ineffective (see [[effective results in number theory]]); this is of particular interest since a major application of this type of result is to bound the number of solutions of some Diophantine equations.