Exponents for three-dimensional simultaneous Diophantine approximations

Moshchevitin, Nikolay. "Exponents for three-dimensional simultaneous Diophantine approximations." Czechoslovak Mathematical Journal 62.1 (2012): 127-137. <http://eudml.org/doc/247147>.

@article{Moshchevitin2012,
abstract = {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 \[ \begin\{aligned\} \alpha (\Theta ) &= \sup \lbrace \gamma >0\colon \limsup \_\{t\rightarrow +\infty \} t^\gamma \psi \_\Theta (t) <+\infty \rbrace ,\\ \beta (\Theta ) &= \sup \lbrace \gamma >0\colon \liminf \_\{t\rightarrow +\infty \} t^\gamma \psi \_\Theta (t)<+\infty \rbrace \end\{aligned\} \] we prove \[ \beta (\Theta ) \ge \frac\{1\}\{2\} \Bigg ( \frac\{\alpha (\Theta )\}\{1-\alpha (\Theta )\} +\sqrt\{\Big (\frac\{\alpha (\Theta )\}\{1-\alpha (\Theta )\} \Big )^2 +\frac\{4\alpha (\Theta )\}\{1-\alpha (\Theta )\}\} \Bigg ) \alpha (\Theta ). \]},
author = {Moshchevitin, Nikolay},
journal = {Czechoslovak Mathematical Journal},
keywords = {Diophantine approximations; Diophantine exponents; Jarník's transference principle; Diophantine approximations; Diophantine exponent},
language = {eng},
number = {1},
pages = {127-137},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Exponents for three-dimensional simultaneous Diophantine approximations},
url = {http://eudml.org/doc/247147},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Moshchevitin, Nikolay
TI - Exponents for three-dimensional simultaneous Diophantine approximations
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 1
SP - 127
EP - 137
AB - 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 \[ \begin{aligned} \alpha (\Theta ) &= \sup \lbrace \gamma >0\colon \limsup _{t\rightarrow +\infty } t^\gamma \psi _\Theta (t) <+\infty \rbrace ,\\ \beta (\Theta ) &= \sup \lbrace \gamma >0\colon \liminf _{t\rightarrow +\infty } t^\gamma \psi _\Theta (t)<+\infty \rbrace \end{aligned} \] we prove \[ \beta (\Theta ) \ge \frac{1}{2} \Bigg ( \frac{\alpha (\Theta )}{1-\alpha (\Theta )} +\sqrt{\Big (\frac{\alpha (\Theta )}{1-\alpha (\Theta )} \Big )^2 +\frac{4\alpha (\Theta )}{1-\alpha (\Theta )}} \Bigg ) \alpha (\Theta ). \]
LA - eng
KW - Diophantine approximations; Diophantine exponents; Jarník's transference principle; Diophantine approximations; Diophantine exponent
UR - http://eudml.org/doc/247147
ER -