5
$\begingroup$

Which theorems from classical theory of continued fractions have 3-(or multi-) dimesional analogs?

Of cause classical one is a periodicity of Klein polyhedra. Probably there are some more...

$\endgroup$
  • 3
    $\begingroup$ You might like the Ph.D. thesis (in Russian, geometrie.tugraz.at/karpenkov ) by Oleg Karpenkov which is all about multidim continued fractions. $\endgroup$ – Igor Pak Nov 8 '12 at 8:39
  • 1
    $\begingroup$ He studied multidimesional Klein model of continued fractions. His thesis contains different theorems concerning this model, but they are not direct analogs of clasical theorem. Though he deeply studied 3-D Gauss-Kuz'min statistics. $\endgroup$ – Alexey Ustinov Nov 8 '12 at 9:36
2
$\begingroup$

(1) There is 3-D analog of Vahlen's theorem, see http://link.springer.com/article/10.1007%2Fs11006-006-0018-6?LI=true

(2) 3-D isolation theorems and extemal Davenport forms (see Cassels "An Introduction to the Geometry of Numbers" and Swinnerton-Dyer, "On the product of three homogeneous linear forms" Acta Arith., 1971, 18, 371-385). The key role here play numbers $2\cos\frac{2\pi}7$, $2\cos\frac{4\pi}7$, $2\cos\frac{6\pi}7$ (3-D Golden Ratios).

$\endgroup$
  • 1
    $\begingroup$ @Alexey, the modern isolation theorem is due to Lindenstrauss and Weiss - "On Sets Invariant under the Action of the Diagonal Group". $\endgroup$ – Asaf Nov 8 '12 at 11:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.