Abstract

Consider a real matrix $\Theta$ consisting of rows $({𝜃}_{i,1},\dots <!--FUNCTION\; APPLICATION-->,{𝜃}_{i,n})$ for $1\le i\le m$. The problem of making the system of linear forms $x_1{𝜃}_{i,1}+\cdots +x_n{𝜃}_{i,n}-y_i$ for integers $x_j$, $y_i$ small naturally induces an ordinary and a uniform exponent of approximation, denoted by $w(\Theta )$ and $ŵ(\Theta )$ respectively. For $m=1$, a sharp lower bound for the ratio $w(\Theta )∕ŵ(\Theta )$ 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 bound applies to general $m>1$, but is probably not optimal in this case. Thereby we also complement a similar conditional result of Moshchevitin, who imposed a different assumption on the best approximations. Our hypothesis is satisfied in particular for $m=1$, $n=2$ and thereby unconditionally confirms a previous observation of Jarník. We formulate our results in a very 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.

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Mathematical Subject Classification 2010

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