Exponents for three-dimensional simultaneous Diophantine approximations

Abstract

Let Θ = (θ ₁,θ ₂,θ ₃) ∈ ℝ³. Suppose that 1, θ ₁, θ ₂, θ ₃ are linearly independent over ℤ. For Diophantine exponents

$$\begin{gathered} \alpha (\Theta ) = sup\left\{ {\gamma > 0: \mathop {\lim }\limits_{t \to } \mathop {\sup }\limits_{ + \infty } t^\gamma \psi _\Theta (t) < + \infty } \right\}, \hfill \\ \beta (\Theta ) = sup\left\{ {\gamma > 0: \mathop {\lim }\limits_{t \to } \mathop {\inf }\limits_{ + \infty } t^\gamma \psi _\Theta (t) < + \infty } \right\} \hfill \\ \end{gathered}$$

we prove

$$\beta (\Theta ) \ge {1 \over 2}\left( {{{\alpha (\Theta )} \over {1 - \alpha (\Theta )}} + \sqrt {{{\left( {{{\alpha (\Theta )} \over {1 - \alpha (\Theta )}}} \right)}^2} + {{4\alpha (\Theta )} \over {1 - \alpha (\Theta )}}} } \right)\alpha (\Theta )$$

.