Geodesic
Fractional-linear invariance of the symplex-module algorithm for decomposition in multidimensional continued fractions
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 33, Tome 458 (2017), pp. 77-103

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Using the simplex-module algorithm one can decompose real numbers $\alpha=(\alpha_1,\dots,\alpha_d)$ into multidimensional continued fractions. We verified the invariance of this algorithm under fractional-linear transformations $\alpha'=(\alpha'_1,\dots,\alpha'_d)=U\langle\alpha\rangle$ with matrices $U$ in the unimodular group $\mathrm{GL}_{d+1}(\mathbb Z)$, and prove the conservation of a linear recurrence and the approximation order for convergent fractions to the transformed $\alpha'$. 
@article{ZNSL_2017_458_a5,
     author = {V. G. Zhuravlev},
     title = {Fractional-linear invariance of the symplex-module algorithm for decomposition in multidimensional continued fractions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {77--103},
     publisher = {mathdoc},
     volume = {458},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_458_a5/}
}
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V. G. Zhuravlev. Fractional-linear invariance of the symplex-module algorithm for decomposition in multidimensional continued fractions. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 33, Tome 458 (2017), pp. 77-103. http://geodesic.mathdoc.fr/item/ZNSL_2017_458_a5/