MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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...

share|cite|improve this question
You might like the Ph.D. thesis (in Russian, ) by Oleg Karpenkov which is all about multidim continued fractions. – Igor Pak Nov 8 '12 at 8:39
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. – Alexey Ustinov Nov 8 '12 at 9:36
up vote 2 down vote accepted

(1) There is 3-D analog of Vahlen's theorem, see

(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).

share|cite|improve this answer
@Alexey, the modern isolation theorem is due to Lindenstrauss and Weiss - "On Sets Invariant under the Action of the Diagonal Group". – Asaf Nov 8 '12 at 11:08
Asaf, thank you for the reference – Alexey Ustinov Nov 12 '12 at 23:17

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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