Applications of Siegel's Lemma to a system of linear forms and its minimal points

Abstract

Consider a real matrix $\Theta$ consisting of rows $({\theta }_{i,1},\dots ,{\theta }_{i,n})$, for $1\le i\le m$. The problem of making the system linear forms $x_1{\theta }_{i,1}+\cdots +x_n{\theta }_{i,n}-y_i$ for integers $x_j,y_i$ small naturally induces an ordinary and a uniform exponent of approximation, denoted by $w(\Theta )$ and $\hat{w}(\Theta )$ respectively. For $m=1$, a sharp lower bound for the ratio $w(\Theta )/\hat{w}(\Theta )$ was recently established by Marnat and Moshchevitin. We give a short, new proof of this result upon a hypothesis on the best approximation integer vectors associated to $\Theta$. Our conditional result extends to general $m>1$ (but may not be optimal in this case). Moreover, our hypothesis is always satisfied in particular for $m=1,n=2$ and thereby unconditionally confirms a previous observation of Jarník. We formulate our results in the more general context of approximation of subspaces of Euclidean spaces by lattices. We further establish criteria upon which a given number $\ell$ of consecutive best approximation vectors are linearly independent. Our method is based on Siegel's Lemma.

29 pages

References in corpus (3)

• On some open problems in Diophantine approximation
• The best Diophantine approximations: the phenomenon of degenerate dimension
• Diophantine exponents for systems of linear forms in two variables

Cited by in corpus (2)

• Geometry of Diophantine exponents
• On Some Properties of Irrational Subspaces