Geodesic
Vectors of Given Diophantine Type
Matematičeskie zametki, Tome 80 (2006) no. 3, pp. 328-338

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For a function $\varphi(y)=o(y^{-1/s})$, $y\to\infty$, we prove the existence of vectors $\overline\alpha\in\mathbb R^s$ admitting, for any $\varepsilon>0$, infinitely many simultaneous $\varphi(1+\varepsilon)$-approximations, but not admitting any simultaneous $\varphi$-approximations.
Keywords: vector of Diophantine type, simultaneous Diophantine approximations, Euclidean space, simultaneous $\varphi$-approximation. 
@article{MZM_2006_80_3_a1,
     author = {R. K. Akhunzhanov and N. G. Moshchevitin},
     title = {Vectors of {Given} {Diophantine} {Type}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {328--338},
     publisher = {mathdoc},
     volume = {80},
     number = {3},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_80_3_a1/}
}
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R. K. Akhunzhanov; N. G. Moshchevitin. Vectors of Given Diophantine Type. Matematičeskie zametki, Tome 80 (2006) no. 3, pp. 328-338. http://geodesic.mathdoc.fr/item/MZM_2006_80_3_a1/