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...



3-D continued fractions
(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).


