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Keywords:
Diophantine approximations; Diophantine exponents; Jarník's transference principle
Diophantine approximations; Diophantine exponents; Jarník's transference principle
Summary:
Let $\Theta = (\theta _1,\theta _2,\theta _3)\in \mathbb {R}^3$. Suppose that $1,\theta _1,\theta _2,\theta _3$ are linearly independent over $\mathbb {Z}$. For Diophantine exponents $$ \begin {aligned} \alpha (\Theta ) &= \sup \{\gamma >0\colon \limsup _{t\to +\infty } t^\gamma \psi _\Theta (t) <+\infty \},\\ \beta (\Theta ) &= \sup \{\gamma >0\colon \liminf _{t\to +\infty } t^\gamma \psi _\Theta (t)<+\infty \} \end {aligned} $$ we prove $$ \beta (\Theta ) \ge \frac {1}{2} \Bigg ( \frac {\alpha (\Theta )}{1-\alpha (\Theta )} +\sqrt {\Big (\frac {\alpha (\Theta )}{1-\alpha (\Theta )} \Big )^2 +\frac {4\alpha (\Theta )}{1-\alpha (\Theta )}} \Bigg ) \alpha (\Theta ). $$
References:
[1] Jarník, V.: Contribution à la théorie des approximations diophantiennes linéaires et homogènes. Czech. Math. J. 4 (1954), 330-353 Russian, French summary. MR 0072183 | Zbl 0057.28303
[2] Laurent, M.: Exponents of Diophantine approximations in dimension two. Can. J. Math. 61 (2009), 165-189. DOI 10.4153/CJM-2009-008-2 | MR 2488454
[3] Moshchevitin, N. G.: Contribution to Vojtěch Jarník. Preprint available at arXiv:0912.2442v3. MR 0095106
[4] Moshchevitin, N. G.: Khintchine's singular Diophantine systems and their applications. Russ. Math. Surv. 65 433-511 (2010), Translation from Uspekhi Mat. Nauk. 65 43-126 (2010). DOI 10.1070/RM2010v065n03ABEH004680 | MR 2682720 | Zbl 1225.11094
[5] Schmidt, W. M.: On heights of algebraic subspaces and Diophantine approximations. Ann. Math. (2) 85 (1967), 430-472. DOI 10.2307/1970352 | MR 0213301 | Zbl 0152.03602
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