Geodesic
Directional Rational Approximations with Some Applications to Algebraic Geometry
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Birational geometry: Linear systems and finitely generated algebras, Tome 240 (2003), pp. 73-81

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We discuss the existence and order of accuracy of directional rational approximations of points in $\mathbb R^n$. As an application, we provide a simplified version of the lemma on Diophantine approximation. We also prove a generalization of the Rationality Theorem of log-MMP on the basis of a newer geometric approach. 
@article{TM_2003_240_a4,
     author = {A. A. Borisov and V. V. Shokurov},
     title = {Directional {Rational} {Approximations} with {Some} {Applications} to {Algebraic} {Geometry}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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     volume = {240},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2003_240_a4/}
}
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A. A. Borisov; V. V. Shokurov. Directional Rational Approximations with Some Applications to Algebraic Geometry. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Birational geometry: Linear systems and finitely generated algebras, Tome 240 (2003), pp. 73-81. http://geodesic.mathdoc.fr/item/TM_2003_240_a4/