It is of a qualitative type, stating that algebraic numbers cannot have many rational approximations that are 'very good'.
Over half a century, the meaning of very good here was refined by a number of mathematicians, starting with Joseph Liouville in 1844 and continuing with work of Axel Thue (1909), Carl Ludwig Siegel (1921), Freeman Dyson (1947), and Klaus Roth (1955).
Roth's theorem states that every irrational algebraic number
The first result in this direction is Liouville's theorem on approximation of algebraic numbers, which gives an approximation exponent of d for an algebraic number α of degree d ≥ 2.
which he applied to prove the finiteness of the solutions of Thue equation.
Roth's result with exponent 2 is in some sense the best possible, because this statement would fail on setting
: by Dirichlet's theorem on diophantine approximation there are infinitely many solutions in this case.
However, there is a stronger conjecture of Serge Lang that can have only finitely many solutions in integers p and q.
[2] The fact that we do not actually know C(ε) means that the project of solving the equation, or bounding the size of the solutions, is out of reach.
The proof technique involves constructing an auxiliary multivariate polynomial in an arbitrarily large number of variables depending upon
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.
There is a higher-dimensional version, Schmidt's subspace theorem, of the basic result.
There are also numerous extensions, for example using the p-adic metric,[3] based on the Roth method.