with C(ε) a positive constant depending only on ε > 0. This cannot be bettered in the sense that setting ε = 0 here meets the case that real numbers x generally do have rational approximations p/q to within q−2. That is Dirichlet's theorem on diophantine approximation. Therefore Roth's result closed the gap, which in the earlier work was still unknown ground. For comparison, the original Thue's theorem from 1909 replaces the exponent −(2 + ε) by −(½d + 1 + ε), where d > 2 is the degree of α.
The proof technique was the construction of an auxiliary function in several variables, leading to a contradiction in the presence of too many good approximations. 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 bounding the number of solutions of some diophantine equations. The fact that we don't actually know C(ε) means that the project of solving the equation, or bounding the size of the solutions, is out of reach. Later work using the methods of Alan Baker made some small impact on effective improvements to Liouville's theorem on diophantine approximation, which gives a bound
(see Liouville number); but the inequalities are still weak.