TITLE: How large (small) can be the measure of a set where a polynomial takes small values ?
QUESTION [7 upvotes]: 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 ?
This, and other interesting variation of this must have been studied in depth. 
I would really appreciate any reference to the relevant literature. 
Also, if there are some interesting variation of this problem I would like to know. 
Thank you.

REPLY [2 votes]: Dear Vagabond,
I think a useful reference containing a lemma in the direction of your question is:
http://www.ams.org/mathscinet-getitem?mr=1652916 (see in particular proposition 3.2)
In this paper, Kleinbock and Margulis show the following result:
$\lambda(\{x\in I: |f(x)|<\varepsilon\})\leq 2k (k+1)^{1/k} (\varepsilon/\|f\|_I)^{1/k} \lambda(I)$
Here $\lambda$ is the Lebesgue measure, $f$ is a polynomial of degree $k$ and $\|f\|_I$ is the supremum of $f$ on $I$.
Best,
Matheus