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 \alpha of degree d\geq2. This is already enough to demonstrate the existence of
transcendental numbers. Thue realised that an exponent less than d would have applications to the solution of
Diophantine equations and in '''Thue's theorem''' from 1909 established an exponent d/2 + 1 + \varepsilon which he applied to prove the finiteness of the solutions of
Thue equations. Siegel's theorem improves this to an exponent about 2\sqrt{d}, and Dyson's theorem of 1947 has exponent about \sqrt{2d}. Roth's result with exponent 2 is in some sense the best possible, because this statement would fail on setting \varepsilon = 0: 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 :\left|\alpha - \frac{p}{q}\right| can have only finitely many solutions in integers
p and
q. If one lets \alpha run over the whole of the set of
real numbers, not just the algebraic reals, then both Roth's conclusion and Lang's hold for
almost all \alpha. So both the theorem and the conjecture assert that a certain
countable set misses a certain set of
measure zero. The theorem is not currently
effective: that is, there is no
bound known on the possible values of p and q given \alpha. showed that Roth's techniques could be used to give an effective bound for the number of p and q satisfying the inequality, using a "gap" principle. The fact that we do not actually know C(\varepsilon) means that the project of solving the equation, or bounding the size of the solutions, is out of reach. == Proof technique ==