Geodesic
Vectors of a given Diophantine type. II
Sbornik. Mathematics, Tome 204 (2013) no. 4, pp. 463-484

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We prove the existence of a family of vectors with continuum many elements $\mathbf v\in\mathbb{R}^s$ admitting infinitely many simultaneous $(\varphi(p)/p^{1/s})(1+B\cdot\varphi^{1+1/s}(p))$-approximations and admitting no simultaneous $(\varphi(p)/p^{1/s})(1-B\cdot\varphi^{1+1/s}(p))$-approximation. We prove that for $0 the closed interval $[t,t(1+16B\cdot t^{1+1/s})]$ contains an element of the $s$-dimensional Lagrange spectrum. Here $A$, $B$ and $T$ stand for some positive constants depending on the dimension $s$ only and $\varphi$ is a positive nonincreasing function of positive integer argument such that $\varphi(1)\le A$. Bibliography: 5 titles.
Keywords: simultaneous Diophantine approximations, Lagrange spectrum, Euclidean space, simultaneous $\psi$-approximation. 
R. K. Akhunzhanov. Vectors of a given Diophantine type. II. Sbornik. Mathematics, Tome 204 (2013) no. 4, pp. 463-484. http://geodesic.mathdoc.fr/item/SM_2013_204_4_a0/
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