An operator-norm version of Siegel's Lemma - MathOverflow 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 An operator-norm version of Siegel's 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>