Thue-Siegel-Roth theorem

The Thue-Siegel-Roth theorem from the theory of Diophantine approximations in number theory was proved by Klaus Friedrich Roth after preliminary work by Axel Thue and Carl Ludwig Siegel in 1955.

It says that for every algebraic number and every the inequality ( p , q relatively prime ) ${\displaystyle \alpha }$ ${\displaystyle \varepsilon >0}$

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|<q^{-(2+\varepsilon )}}$		(Inequality 1)

only has a finite number of solutions. By leaving these finitely many solutions aside, one can deduce from (Inequality 1) that for sufficiently large q for every irrational one holds: ${\displaystyle \alpha }$

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|>C(\varepsilon ,\alpha )q^{-(2+\varepsilon )}}$		(Inequality 2)

with a C dependent only on and . This is the form in which Tue-Siegel-Roth's sentence is usually presented. This is the “best” possible such theorem, since according to Peter Gustav Lejeune Dirichlet ( Dirichlet's approximation theorem ) every real number has approximants p / q that are closer than . There are even infinitely many, e.g. B. the approximants of the continued fraction representations of these numbers (whose special role the theorem thus also shows). That is, for every irrational number there are infinitely many rational numbers with such that: ${\displaystyle \varepsilon }$ ${\displaystyle \alpha }$ ${\displaystyle \alpha }$ ${\displaystyle q^{-2}}$ ${\displaystyle \alpha }$ ${\displaystyle {\frac {p}{q}}}$ ${\displaystyle q>0}$

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|<q^{-2}}$

After that, upper bounds for exponents were gradually determined, so that there are finitely many rational approximate solutions for algebraic irrational numbers with ${\displaystyle \mu }$ ${\displaystyle {\frac {p}{q}}}$ ${\displaystyle \alpha }$

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|<q^{-\mu }}$		(Inequality 3)

gives. Joseph Liouville showed in 1844 , with (see Diophantine approximation ). Here n is the degree of the algebraic equation with the root . Elementary considerations also show that is (see above). This was known and refined limits were sought. Axel Thue showed in 1908 that and Carl Ludwig Siegel in his dissertation in 1921 (whereby he communicated the result to his teacher Frobenius as early as 1916) that . Roth showed that 2 is actually the optimal bound, because there are only finitely many solutions for. ${\displaystyle \mu <n}$ ${\displaystyle n>1}$ ${\displaystyle \alpha }$ ${\displaystyle \mu >2}$ ${\displaystyle 2<\mu \leq n}$ ${\displaystyle \mu \leq n/2+1}$ ${\displaystyle \mu \leq 2{\sqrt {n}}}$ ${\displaystyle \mu \leq 2+\varepsilon }$

The proof of the theorem is extensive and can be found, for example, in the textbooks of Theodor Schneider or John Cassels .

Roth's proof does not give a method to find such solutions or to restrict C. This would be interesting to learn about the number of solutions to Diophantine equations (i.e., integer or rational solutions to algebraic equations for which, for example, that in (Inequality 2) is a real root). Such effective methods were introduced to the theory of transcendent numbers and Diophantine equations by Alan Baker in the 1960s . Thue-Siegel-Roth's theorem also follows from Wolfgang Schmidt's subspace theorem . This also gave a generalization for the simultaneous approximation of several algebraic numbers . Let be linearly independent over the rational numbers and any positive real number, then there are only a finite number of n-tuples of rational numbers with ${\displaystyle \alpha }$ ${\displaystyle x_{1},\cdots ,x_{n}}$ ${\displaystyle 1,x_{1},\cdots ,x_{n}}$ ${\displaystyle \varepsilon }$ ${\displaystyle {\frac {p_{1}}{q}},\cdots ,{\frac {p_{n}}{q}}}$

${\displaystyle |x_{i}-{\frac {p_{i}}{q}}|<q^{-(1+{\frac {1}{n}}+\varepsilon )},\quad i=1,\ldots ,n.}$

There is also a p-adic version of the Thue-Siegel-Roth theorem.

Applying the Thue-Siegel-Roth theorem one can find new transcendent numbers. The set of Liouville delivered these in the form Liouvillescher numbers . With the Thue-Siegel-Roth theorem one only needs to find irrational numbers that can be approximated better than by rational numbers and not as in Liouville's theorem. An example is the evidence of transcendence for the number ${\displaystyle {\frac {1}{q^{2}}}}$ ${\displaystyle {\frac {1}{q^{n}}}}$

${\displaystyle \tau =1234567891011121314\cdots }$

So the number that arises when you write all decimal numbers one after the other. The same applies if the number is not based on the decimal system, but rather based on the manipulated variable system based on base 3. The original proof is from Kurt Mahler (1946) and the proof does not necessarily require Thue-Siegel-Roth's Theorem. is not a Liouville number. ${\displaystyle \tau }$

literature

• Theodor Schneider: Introduction to the transcendent numbers, Springer 1957
• John Cassels: An introduction to diophantine approximation, Cambridge UP 1957
• William LeVeque : Topics in number theory, Volume 2, 1956, Chapter 4, reprinted in Dover 2002

Individual evidence

1. ↑ Klaus Friedrich Roth : Rational approximations to algebraic numbers and Corrigendum. In: Mathematika. Vol. 2, 1955, ISSN  0025-5793 , pp. 1-20 and 168.
2. ↑ Fridtjof Tönniessen, The secret of transcendent numbers, Spektrum Akademischer Verlag 2010, p. 421
3. ↑ Fridtjof Tönniessen, The secret of the transcendent numbers, Spektrum Akademischer Verlag 2010, p. 418
4. ^ Schneider, Introduction to the transcendent numbers, Springer 1957
5. ^ Cassels, An introduction to diophantine approximation, Cambridge UP 1957
6. ^ Proved by D. Ridout, The p-adic generalization of the Thue-Siegel-Roth theorem, Mathematika, Volume 5, 1958, pp. 40-48
7. ↑ Fridtjof Tönniessen, The secret of transcendent numbers, Spektrum Akademischer Verlag 2010, p. 420

Web links

• Ishak The Thue-Siegel-Roth Theorem with the proof from Leveque Topics' book in Number Theory 1956, PDF file