most recent 30 from http://mathoverflow.net 2013-05-19T04:37:58Z http://mathoverflow.net/feeds/question/48394 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/48394/an-operator-norm-version-of-siegels-lemma Seva 2010-12-05T21:18:59Z 2010-12-06T09:59:50Z

<p>Is there a kind of Siegel's Lemma saying that if $M$ is a ``small-height'' integer matrix, then there is a "small-height" vector $x$ with $\|Mx\|=\|M\|\|x\|$? (Here $\|Mx\|$ and $\|x\|$ denote the Euclidean norms, and $\|M\|$ is the operator norm, induced by the Euclidean norm.) </p> <p>I am particularly interested in the case where the elements of $M$ are restricted to the values $0$ and $1$; what can be said in this situation about the vectors $x$ with $\|Mx\|=\|M\|\|x\|$ (or with the weaker property that, say, $\|Mx\|\ge 0.1\|M\|\|x\|$)? Can one guarantee that some of these vectors have, in some reasonable sense, a low height?</p>