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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...

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    $\begingroup$ 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
    Commented Nov 8, 2012 at 8:39
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    $\begingroup$ 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
    Commented Nov 8, 2012 at 9:36

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(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).

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    $\begingroup$ @Alexey, the modern isolation theorem is due to Lindenstrauss and Weiss - "On Sets Invariant under the Action of the Diagonal Group". $\endgroup$
    – Asaf
    Commented Nov 8, 2012 at 11:08

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