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. 
@article{SM_2013_204_4_a0,
     author = {R. K. Akhunzhanov},
     title = {Vectors of a~given {Diophantine} {type.~II}},
     journal = {Sbornik. Mathematics},
     pages = {463--484},
     publisher = {mathdoc},
     volume = {204},
     number = {4},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2013_204_4_a0/}
}
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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/