Geodesic
Unimodular invariance of karyon decompositions of algebraic numbers in multidimensional continued fractions
Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 1, Tome 469 (2018), pp. 96-137

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By the differentiation method of induced toric tilins we find periodic expansions for algebraic irrationalities in multidimensional continued fractions. These expansions give the best karyon approximations with respect to polyhedral norms. The above irrationalities are obtained by the composition of backward continued fraction mappings and unimodular transformations of algebraic units that decompose into a purely periodic continued fractions. The artifact of this expansion several invariants has become: recurrence relations for numerators and denominators of convergent fractions and the rate of multidimensional approximation of irrationalities by rational numbers. 
@article{ZNSL_2018_469_a3,
     author = {V. G. Zhuravlev},
     title = {Unimodular invariance of karyon decompositions of algebraic numbers in multidimensional continued fractions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {96--137},
     publisher = {mathdoc},
     volume = {469},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2018_469_a3/}
}
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V. G. Zhuravlev. Unimodular invariance of karyon decompositions of algebraic numbers in multidimensional continued fractions. Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 1, Tome 469 (2018), pp. 96-137. http://geodesic.mathdoc.fr/item/ZNSL_2018_469_a3/