Nazarov proved a version of Turan's lemma here, which is now known by the name Nazarov-Turan's lemma. A special case of this result is the following.

There is $A>0$ such that for every interval $I \subset \mathbb{R}$ and every measurable set $E \subset I$, the following holds: for every $p(t)=\sum_{k=1}^{n} c_k e^{ i \lambda_kt}$, where $c_k \in \mathbb{C}$ and $\lambda_k \in \mathbb{R}$, we have

$$ \int_{I} |p(t)|^2 dt \leq \left(\frac{A|I|}{|E|}\right)^{2(n-1)} \int_{E} |p(t)|^2 dt. $$

My question is: can we expect to have a similar result as the one above for algebraic polynomials?

Thanks!

Nazarov proved a version of Turán's lemma in Complete Version of Turan’s Lemma for Trigonometric Polynomials on the Unit Circumference, which is now known by the name Nazarov–Turán's lemma. A special case of this result is the following.

There is $A>0$ such that for every interval $I \subset \mathbb{R}$ and every measurable set $E \subset I$, the following holds: for every $p(t)=\sum_{k=1}^{n} c_k e^{ i \lambda_kt}$, where $c_k \in \mathbb{C}$ and $\lambda_k \in \mathbb{R}$, we have

$$ \int_{I} |p(t)|^2 dt \leq \left(\frac{A|I|}{|E|}\right)^{2(n-1)} \int_{E} |p(t)|^2 dt. $$

My question is: can we expect to have a similar result as the one above for algebraic polynomials?