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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(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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