Hurwitz theorem (number theory)

The mathematics knows a number of sets which the name of Adolf Hurwitz are linked. The Hurwitz's theorem of number theory concerns the so-called diophantine approximations of irrational numbers , so the approximation of irrational numbers by fractions . The sentence specifies an upper limit for the quality of the approximation.

The sentence

The sentence can be formulated as follows:

For every irrational number there are infinitely many fully abbreviated fractions , which ${\displaystyle \alpha }$ ${\displaystyle {\tfrac {p}{q}}}$

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{{\sqrt {5}}\cdot q^{2}}}}$

fulfill.

In the proof of the theorem developed by Scheid, properties of the Farey sequences are used in a decisive way .

Goodness of the ceiling

The constant is sharp, so in general it cannot be replaced by a better constant. This can be proven using the irrational number (known in connection with the golden ratio ). ${\displaystyle {\sqrt {5}}}$ ${\displaystyle {\mathcal {\alpha }}=(1+{\sqrt {5}})/2}$

For a single number there may be better approximations, e.g. B. for Liouville numbers . If it is an algebraic number , the exponent of cannot be improved according to the Thue-Siegel-Roth theorem . ${\displaystyle \alpha }$ ${\displaystyle \alpha }$ ${\displaystyle q}$

Related results

• Dirichletscher approximation theorem

literature

• Adolf Hurwitz: About the approximate representation of irrational numbers through rational fractions . In: Math. Ann. , 39, 1891, pp. 279-284
• Harald Scheid : Number Theory . 3. Edition. Spectrum Academic Publishing House, Heidelberg (among others) 2003, ISBN 3-8274-1365-6 .
• Jurjen Ferdinand Koksma : Diophantine approximations . 3. Edition. Springer-Verlag, Berlin 1936.
• William Judson LeVeque : Fundamentals of Number Theory . Addison-Wesley, Reading MA 1977, ISBN 0-201-04287-8 .

Individual evidence

1. ↑ Harald Scheid: Number theory . 3. Edition. Spectrum Akademischer Verlag, Heidelberg (among others) 2003, ISBN 3-8274-1365-6 , p. 64 .
2. ↑ Harald Scheid: Number theory . 3. Edition. Spectrum Akademischer Verlag, Heidelberg (among others) 2003, ISBN 3-8274-1365-6 , p. 64-65 .
3. ↑ Harald Scheid: Number theory . 3. Edition. Spectrum Akademischer Verlag, Heidelberg (among others) 2003, ISBN 3-8274-1365-6 , p. 65 .