Geodesic
Exponents for three-dimensional simultaneous Diophantine approximations
Czechoslovak Mathematical Journal, Tome 62 (2012) no. 1, pp. 127-137
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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 $$ \beta (\Theta ) \ge \frac {1}{2} \Bigg ( \frac {\alpha (\Theta )}{1-\alpha (\Theta )} +\sqrt {\Big (\frac {\alpha (\Theta )}{1-\alpha (\Theta )} \Big )^2 +\frac {4\alpha (\Theta )}{1-\alpha (\Theta )}} \Bigg ) \alpha (\Theta ). $$
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 $$ \beta (\Theta ) \ge \frac {1}{2} \Bigg ( \frac {\alpha (\Theta )}{1-\alpha (\Theta )} +\sqrt {\Big (\frac {\alpha (\Theta )}{1-\alpha (\Theta )} \Big )^2 +\frac {4\alpha (\Theta )}{1-\alpha (\Theta )}} \Bigg ) \alpha (\Theta ). $$
DOI : 10.1007/s10587-012-0001-1
Classification : 11J13
Keywords: Diophantine approximations; Diophantine exponents; Jarník's transference principle 
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Moshchevitin, Nikolay. Exponents for three-dimensional simultaneous Diophantine approximations. Czechoslovak Mathematical Journal, Tome 62 (2012) no. 1, pp. 127-137. doi: 10.1007/s10587-012-0001-1

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