Diophantine exponents for systems of linear forms in two variables

Abstract

We improve on Jarn\'ık'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\'ık (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 β\ge 1/2(α^2-α+1+\sqrt{(α^2-α+1)^2 +4α^2(α-1)}) if 1\le α\le 2 1/2(α^2-1+\sqrt{(α^2-1)^2+4α(α-1)}) if α\ge 2

15 pages

Cited by in corpus (1)

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