How large (small) can be the measure of a set where a polynomial takes small values ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T11:35:24Z http://mathoverflow.net/feeds/question/32815 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/32815/how-large-small-can-be-the-measure-of-a-set-where-a-polynomial-takes-small-valu How large (small) can be the measure of a set where a polynomial takes small values ? Vagabond 2010-07-21T16:57:53Z 2010-07-22T08:36:22Z <p>A $n$-th degree polynomial has precisely $n$ roots. So it is natural to ask the question how large ( and small) can be the measure of a set where a polynomial takes small values ?</p> <p>This, and other interesting variation of this must have been studied in depth. </p> <p>I would really appreciate any reference to the relevant literature. </p> <p>Also, if there are some interesting variation of this problem I would like to know. </p> <p>Thank you.</p> http://mathoverflow.net/questions/32815/how-large-small-can-be-the-measure-of-a-set-where-a-polynomial-takes-small-valu/32818#32818 Answer by Helge for How large (small) can be the measure of a set where a polynomial takes small values ? Helge 2010-07-21T17:10:31Z 2010-07-22T08:36:22Z <p>There is first Polya's estimate that if $f$ is a monic polynomial, then $$|\{x\in \mathbb{R}:\quad |f(x)|\leq 2\}| \leq 4.$$ A proof can be found in the book "Proofs from the book". One can obtain inequalities for non-monic polynomials by rescaling.</p> <p>Second there is Cartan's lemma or estimate. It can for example be found in Levin's book on entire functions. The estimate even holds for analytic functions.</p> <p>The basic statement is:</p> <p>Let $f: G \to \mathbb{C}$ be analytic and assume that $f$ is bounded by $1$ on a disc of radius $2$. Then there are constant $C, c > 0$ such that $$|\{z\in \mathbb{C}:\quad |z| &lt; 1, | f(z)| \leq e^{-s}\}| \leq C \exp\left( - \frac{c}{\log(\varepsilon^{-1})} s \right)$$ where $\varepsilon = |f(0)|$. In fact, this is sharper, since it provides some information on how the set looks. For a polynomial it's just the union of its degree many disks. (For analytic functions countably many).</p> http://mathoverflow.net/questions/32815/how-large-small-can-be-the-measure-of-a-set-where-a-polynomial-takes-small-valu/32822#32822 Answer by Matheus for How large (small) can be the measure of a set where a polynomial takes small values ? Matheus 2010-07-21T17:19:04Z 2010-07-21T17:19:04Z <p>Dear Vagabond,</p> <p>I think a useful reference containing a lemma in the direction of your question is:</p> <p><a href="http://www.ams.org/mathscinet-getitem?mr=1652916" rel="nofollow">http://www.ams.org/mathscinet-getitem?mr=1652916</a> (see in particular proposition 3.2)</p> <p>In this paper, Kleinbock and Margulis show the following result:</p> <p>$\lambda({x\in I: |f(x)|&lt;\varepsilon})\leq 2k (k+1)^{1/k} (\varepsilon/\|f\|_I)^{1/k} \lambda(I)$</p> <p>Here $\lambda$ is the Lebesgue measure, $f$ is a polynomial of degree $k$ and $\|f\|_I$ is the supremum of $f$ on $I$.</p> <p>Best,</p> <p>Matheus</p>