Hurwitz's theorem (number theory)

Mathematics knows a number of sets which are associated with the name of Adolf Hurwitz. The set of Hurwitz number theory concerns the so-called Diophantine approximation of irrational numbers, so the approximation of irrational numbers by fractions. The sentence specifies a limit on the quality of the approximation.

The set

The theorem can be formulated as follows:

For each irrational number exist infinitely many fully shortened fractures, which

. meet

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

Quality of the upper limit

The constant is sharp, so in general not be replaced by a better constant. This can be demonstrated by means of the irrational number ( known in connection with the golden section ).

For a single number, it can give better approximations, eg for Liouville numbers. Is an algebraic number, the exponent of leaves by the theorem of Thue -Siegel -Roth but did not improve.

Related results

• Dirichlet's approximation theorem