Geodesic
Fractional-linear invariance of multidimensional continued fractions
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 33, Tome 458 (2017), pp. 42-76

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With the help of the simplex-karyon algorithm it is possible to decompose real numbers $\alpha=(\alpha_1,\dots,\alpha_d)$ into multidimensional continued fractions. We prove the invariance of this algorithm under fractional-linear transformations $\alpha'=(\alpha'_1,\dots,\alpha'_d)=U\langle\alpha\rangle$ with matrices $U$ from the unimodular group $\mathrm{GL}_{d+1}(\mathbb Z)$. The best convergent fractions of the transformed $\alpha'$ are found. 
@article{ZNSL_2017_458_a4,
     author = {V. G. Zhuravlev},
     title = {Fractional-linear invariance of multidimensional continued fractions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {42--76},
     publisher = {mathdoc},
     volume = {458},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_458_a4/}
}
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V. G. Zhuravlev. Fractional-linear invariance of multidimensional continued fractions. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 33, Tome 458 (2017), pp. 42-76. http://geodesic.mathdoc.fr/item/ZNSL_2017_458_a4/