Exponents for three-dimensional simultaneous Diophantine approximations

Title:

Exponents for three-dimensional simultaneous Diophantine approximations

Creator:

Moshchevitin, Nikolay

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:118bf98e-9065-44c3-9292-984f62d30628
uuid:118bf98e-9065-44c3-9292-984f62d30628

Subject:

Diophantine approximations, Diophantine exponents, and Jarník's transference principle

Type:

model:article and TEXT

Format:

bez média and svazek

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 $$ \beta (\Theta ) \ge \frac {1}{2} \Bigg ( \frac {\alpha (\Theta )}{1-\alpha (\Theta )} +\sqrt {\Big (\frac {\alpha (\Theta )}{1-\alpha (\Theta )} \Big )^2 +\frac {4\alpha (\Theta )}{1-\alpha (\Theta )}} \Bigg ) \alpha (\Theta ). $$.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

127-137

Source:

Czechoslovak Mathematical Journal | 2012 Volume:62 | Number:1

Harvested from:

CDK

Metadata only:

false