Geodesic
Badly approximable vectors in affine subspaces: Jarn\'{\i}k-type result
Čebyševskij sbornik, Tome 12 (2011) no. 2, pp. 77-84.

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Consider irrational affine subspace $ A\subset \mathbb{R}^d$ of dimension $a$. We prove that the set $$ \{\xi =(\xi_1,...,\xi_d) \in { A}:\quad q^{1/a}\cdot \max_{1\le i \le d} ||q\xi_i|| \to \infty,\quad q\to \infty\} $$ is an $\alpha$-winning set for every $\alpha \in (0,1/2]$. 
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     author = {Nikolay Moshchevitin},
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Nikolay Moshchevitin. Badly approximable vectors in affine subspaces: Jarn\'{\i}k-type result. Čebyševskij sbornik, Tome 12 (2011) no. 2, pp. 77-84. http://geodesic.mathdoc.fr/item/CHEB_2011_12_2_a9/

[1] V. Jarník, “Zum Khintchineschen “Übertragungssatz””, Travaux de L'Institut Mathematique de Tbilissi, v. 3, 1938, 193–212 | Zbl

[2] V. Jarník, “Eine Bemerkung uber diophantische approximationen”, Math. Zeitschr., 72 (1959), 187–191 | DOI | MR | Zbl

[3] A. Y. Khinchine, “Über eine klasse linear Diophantine Approximationen”, Rendiconti Circ. Math. Palermo, 50 (1926), 170–195 | DOI

[4] D. Kleinbock, “Extremal supspaces and their submanifolds”, GAFA, 13:2 (2003), 437–466 | DOI | MR | Zbl

[5] D. Kleinbock, “An extension of quantitative nondivergence and applications to Diophantine exponents”, Trans. Amer. Math. Soc., 360 (2008), 6497–6523 | DOI | MR | Zbl

[6] J. Lesca, “Sur un résultat de Jarník”, Acta Arith., 11 (1966), 359–364 | MR | Zbl

[7] N. G. Moshchevitin, “Khintchine's singular Diophantine systems and their applications”, Russian Mathematical Surveys, 65:3 (2010), 43–126 | DOI | MR | Zbl

[8] N. G. Moshchevitin, “Best simultaneous approximations: Norms, signatures, and asymptotic directions”, Math. Notes, 67:5 (2000), 618–624 | DOI | MR | Zbl

[9] N. Moshchevitin, “On Kleinbock's Diophantine result”, Proc. Math. Debrecen, 2009, 2011 (to appear) , arXiv: 0906.1541 | MR | Zbl

[10] W. M. Schmidt, “On badly approximable numbers and certain games”, Trans. Amer. Math. Soc., 123 (1966), 178–199 | DOI | MR | Zbl

[11] W. M. Schmidt, Diophantine Approximations, Lect. Not. Math., 785, 1980 | Zbl

[12] Y. Zhang, “Diophantine exponents of affine subspaces: The simultaneous approximation case”, J. Number Theory, 129:8 (2009), 1976–1989 | DOI | MR | Zbl