Exponents for three-dimensional simultaneous Diophantine approximations

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 $$\alpha (\Theta )=\sup \{\gamma >0:\underset{t\to +\infty }{\mathrm{limsup}}{t}^{\gamma }{\psi }_{\Theta }(t)<+\infty \},$$ $$\beta (\Theta )=\sup \{\gamma >0:\underset{t\to +\infty }{\mathrm{liminf}}{t}^{\gamma }{\psi }_{\Theta }(t)<+\infty \}$$ we prove $$\beta (\Theta )\ge 1/2(\alpha (\Theta )/1-\alpha (\Theta )+\sqrt{\alpha (\Theta )/1-\alpha (\Theta ){)}^2+4\alpha (\Theta )/1-\alpha (\Theta )})\alpha (\Theta )$$

8 pages, correction of misprints, submitted to Czechoslovak Mathematical Journal

References in corpus (2)

• Contribution to Vojtech Jarnik
• Exponents of Diophantine Approximation in dimension two