- Thue–Siegel–Roth theorem
In

mathematics , the**Thue–Siegel–Roth theorem**, also known simply as**Roth's theorem**, is a foundational result indiophantine approximation toalgebraic number s. It is of a qualitative type, stating that a given algebraic number α may not have too manyrational number approximations, that are 'very good'. Over half a century, the meaning of "very good" here was refined by a number of mathematicians, starting withAxel Thue , and continuing with work ofCarl 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,*] It states that for given ε > 0, the inequality**2**, pages 1-20 and 168 (1955):$left|alpha\; -\; frac\{p\}\{q\}\; ight|\; <\; q^\{-(2\; +\; epsilon)\}$

can have only finitely many solutions in

coprime integers "p" and "q". Therefore, by taking aninfimum , we can assert that any irrational α satisfies:$left|alpha\; -\; frac\{p\}\{q\}\; ight|\; >\; C(epsilon)q^\{-(2\; +\; epsilon)\}$

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 isDirichlet'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 function in several variables, leading to a contradiction in the presence of too many good approximations. By its nature, it was ineffective (seeeffective 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 somediophantine equation s. 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 ofAlan Baker made some small impact on "effective" improvements toLiouville's theorem on diophantine approximation , which gives a bound:$left|alpha-\{p\; over\; q\}\; ight|\; geq\; Cq^\{-d\}$

(see

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,*] , based on the Roth method.**5**, pages 40-48, (1958)**ee also**Granville-Langevin conjecture **References**

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