Geodesic
THE LAGRANGE THEOREM FOR MULTIDIMENSIONAL DIOPHANTINE APPROXIMATION
Acta mathematica Universitatis Comenianae, Tome 67 (1998) no. 2 
P. Meignen. THE LAGRANGE THEOREM FOR MULTIDIMENSIONAL DIOPHANTINE APPROXIMATION. Acta mathematica Universitatis Comenianae, Tome 67 (1998) no. 2. http://geodesic.mathdoc.fr/item/AMUC_1998_67_2_a2/
@article{AMUC_1998_67_2_a2,
     author = {P. Meignen},
     title = {THE {LAGRANGE} {THEOREM} {FOR} {MULTIDIMENSIONAL} {DIOPHANTINE} {APPROXIMATION}},
     journal = {Acta mathematica Universitatis Comenianae},
     year = {1998},
     volume = {67},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/AMUC_1998_67_2_a2/}
}
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In this paper we give a necessary and sufficient condition for $z$ in the floor of the Poincare half-space to have periodicity in the multidimensional Diophantine approximation by convergents using the Hermite algorithm. We examine in detail the structure of the corresponding sequences and give some examples