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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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You might like the Ph.D. thesis (in Russian, geometrie.tugraz.at/karpenkov ) by Oleg Karpenkov which is all about multidim continued fractions. –  Igor Pak Nov 8 '12 at 8:39
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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

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up vote 2 down vote accepted

(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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@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

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