Exponents of Diophantine Approximation in dimension two

TL;DR

This paper introduces a new framework of four exponents to measure Diophantine approximation in two dimensions, establishing their relationships and showing their realizability for any quadruple satisfying certain inequalities.

Contribution

It defines a quadruple of exponents for 2D Diophantine approximation, refines existing inequalities, and proves the existence of points realizing any admissible quadruple.

Findings

The four exponents satisfy specific inequalities refining Khintchine's transference principle.

A relation between the two uniform exponents is established by Jarník's equation.

Any quadruple satisfying the inequalities can be realized by some point in R^2.

Abstract

Let $\Theta=(\alpha,\beta)$ be a point in $\bR^2$, with $1,\alpha,\beta$ linearly independent over $\bQ$. We attach to $\Theta$ a quadruple $\Omega(\Theta)$ of exponents which measure the quality of approximation to $\Theta$ both by rational points and by rational lines. The two ``uniform'' components of $\Omega(\Theta)$ are related by an equation, due to Jarn{\'\i}k, and the four exponents satisfy two inequalities which refine Khintchine's transference principle. Conversely, we show that for any quadruple $\Omega$ fulfilling these necessary conditions, there exists a point $\Theta\in \bR^2$ for which $\Omega(\Theta) =\Omega$.