Thue-Siegel-Roth theorem
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In mathematics, the Thue-Siegel-Roth theorem, also known simply as Roth's theorem, is a foundational result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that a given algebraic number α may not have too many rational number approximations, that are 'very good'. Over half a century, the meaning of very good here was refined by a number of mathematicians, starting with Axel Thue, and continuing with work of Carl Ludwig Siegel. Roth's result, which is best possible of its kind, dates from 1955. It states that for given ε > 0, the inequality
- <math>|\alpha - \frac{p}{q}| < q^{-(2 + \epsilon)}<math>
can have only finitely many solutions in coprime integers p and q. Therefore, by taking an infimum, we can assert that any irrational α satisfies
- <math>|\alpha - \frac{p}{q}| > C(\epsilon)q^{-(2 + \epsilon)}<math>
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 approximation. 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
- <math>|\alpha-{p \over q}| \geq Cq^{-d}<math>
(see Liouville number); but the inequalities are still weak.
There is a higher-dimensional version, Schmidt's theorem, of the basic result. There are also numerous extensions, for example using the p-adic metric, based on the Roth method.