Exponents for three-dimensional simultaneous Diophantine approximations

TL;DR

This paper establishes a new lower bound relationship between the Diophantine exponents () and () for three-dimensional simultaneous approximation, advancing understanding of their interplay.

Contribution

It provides the first explicit inequality linking () and () in three dimensions, improving bounds in Diophantine approximation theory.

Findings

Derived a new lower bound for () in terms of ()

Established the inequality involving () and () for  in \u211d^3

Enhanced theoretical understanding of Diophantine exponents in three dimensions.

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 $$ \alpha(\Theta) = \sup \{\gamma >0:\,\,\, \limsup_{t\to +\infty} t^\gamma \psi_\Theta (t) <+\infty \} ,$$ $$\beta(\Theta) = \sup \{\gamma >0:\,\,\, \liminf_{t\to +\infty} t^\gamma \psi_\Theta (t) <+\infty\} $$ we prove $$ \beta (\Theta) \ge {1/2} ({\alpha (\Theta)}/{1-\alpha(\Theta)} +\sqrt{{\alpha(\Theta)}/{1-\alpha(\Theta)})^2 +{4\alpha(\Theta)}/{1-\alpha(\Theta)}}) \alpha (\Theta) $$