Kronecker's approximation theorem

The approximation of Kronecker one of the many theorems of mathematics , which named the German mathematician Leopold Kronecker are connected. This theorem is on an equal footing with other well-known approximation theorems from the field of Diophantine approximation such as Liouville's approximation theorem , Dirichlet's approximation theorem or Hurwitz's theorem of number theory . Like those, Kronecker's approximation theorem deals with the problem of the approximation of irrational numbers by fractions .

Formulation of the sentence

The sentence can be formulated as follows:

Given are real numbers     and     with     and furthermore a natural number  . ${\displaystyle \eta }$ ${\displaystyle \delta }$ ${\displaystyle \delta >0}$   ${\displaystyle n}$

Then for every irrational number   there exist natural numbers   and     with    such that ${\displaystyle \xi }$     ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle q>n}$

${\displaystyle \left|\xi -{\frac {p+{\eta }}{q}}\right|<{\frac {{\frac {1}{2}}+{\frac {1}{\sqrt {5}}}+\delta }{q^{2}}}}$

is satisfied.

In particular, for every irrational number is     the set ${\displaystyle \xi }$

${\displaystyle \{{q\cdot \xi }-\lfloor {q\cdot \xi }\rfloor \mid q\in \mathbb {N} \}}$

close in the open unit interval    . ${\displaystyle {]0,1[}}$

comment

The theorem can be inferred as a direct consequence of the Hurwitz theorem of number theory and can thus be viewed as a consequence of the special properties of the Farey sequences .

literature

• Jurjen Ferdinand Koksma : Diophantine approximations . Springer-Verlag , Berlin et al. 1974, ISBN 3-540-06300-5 (reprint of the 1936 edition).
• Georg Johann Rieger : Number Theory . Vandenhoeck & Ruprecht , Göttingen 1976, ISBN 3-525-40138-8 .
• Harald Scheid : Number Theory . 3. Edition. Spektrum Akademischer Verlag , Heidelberg et al. 2003, ISBN 3-8274-1365-6 .

Individual evidence

1. ↑ Koksma: Diophantine Approximations . 1974, p. 83 .
2. ^ Scheid: Number theory . 2003, p. 66 .
3. ↑ Rieger: Number Theory . 1976, p. 139 .
4. ↑ = integer function of . ${\displaystyle \lfloor x\rfloor }$ ${\displaystyle x}$
5. ^ Scheid: Number theory . 2003, p. 62 ff .