- Thue–Siegel–Roth theorem
mathematics, the Thue–Siegel–Roth theorem, also known simply as Roth's theorem, is a foundational result in diophantine approximationto algebraic numbers. It is of a qualitative type, stating that a given algebraic number α may not have too many rational numberapproximations, 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. Klaus Roth's result, which is best possible of its kind, dates from 1955. [K. F. Roth, "Rational approximations to algebraic numbers" and "Corrigendum", Mathematika, 2, pages 1-20 and 168 (1955)] It states that for given ε > 0, the inequality
can have only finitely many solutions in
coprimeintegers "p" and "q". Therefore, by taking an infimum, we can assert that any irrational α satisfies
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 −(½"d" + 1 + ε), where "d" > 2 is the degree of α.
The proof technique was the construction of an
auxiliary functionin 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 Bakermade some small impact on "effective" improvements to Liouville's theorem on diophantine approximation, which gives a bound
Liouville number); but the inequalities are still weak.
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[D. Ridout, "The p-adic generalization of the Thue-Siegel-Roth theorem", Mathematika, 5, pages 40-48, (1958)] , based on the Roth method.
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