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arXiv · 1209.1697

Diophantine exponents for systems of linear forms in two variables

Nikolay G. Moshchevitin
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Abstract

We improve on Jarn\'ık's inequality between uniform Diophantine exponent $α$ and ordinary Diophantine exponent $β$ for a system of $ n\ge 2$ real linear forms in two integer variables. Jarn\'ık (1949, 1954) proved that $β\ge α(α-1)$. In the present paper we give a better bound in the case $α>1$. 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

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Nikolay G. Moshchevitin. 2012-09-08. Diophantine exponents for systems of linear forms in two variables. https://arxiv.org/abs/1209.1697

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