Diophantine exponents for systems of linear forms in two variables

TL;DR

This paper refines inequalities relating uniform and ordinary Diophantine exponents for systems of two-variable linear forms, providing sharper bounds for exponents when lpha > 1.

Contribution

It improves Jarnedk's inequality bounds for Diophantine exponents in systems of linear forms in two variables, offering explicit formulas for better bounds when lpha > 1.

Findings

Derived new bounds for eta in terms of lpha for lpha > 1.

Provided explicit formulas for eta depending on lpha in different ranges.

Enhanced understanding of Diophantine approximation inequalities for systems of linear forms.

Abstract

We improve on Jarn\'{\i}k's inequality between uniform Diophantine exponent $\alpha $ and ordinary Diophantine exponent $\beta$ for a system of $ n\ge 2$ real linear forms in two integer variables. Jarn\'{\i}k (1949, 1954) proved that $\beta \ge \alpha (\alpha -1)$. In the present paper we give a better bound in the case $\alpha >1$. We prove that \beta \ge 1/2(\alpha^2-\alpha+1+\sqrt{(\alpha^2-\alpha+1)^2 +4\alpha^2(\alpha-1)}) if 1\le \alpha \le 2 1/2(\alpha^2-1+\sqrt{(\alpha^2-1)^2+4\alpha (\alpha-1)}) if \alpha \ge 2