1. Exponents for three-dimensional simultaneous Diophantine approximations
- Creator:
- Moshchevitin, Nikolay
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- Diophantine approximations, Diophantine exponents, and Jarník's transference principle
- Language:
- English
- Description:
- 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 $$ \beta (\Theta ) \ge \frac {1}{2} \Bigg ( \frac {\alpha (\Theta )}{1-\alpha (\Theta )} +\sqrt {\Big (\frac {\alpha (\Theta )}{1-\alpha (\Theta )} \Big )^2 +\frac {4\alpha (\Theta )}{1-\alpha (\Theta )}} \Bigg ) \alpha (\Theta ). $$.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public