Let $ p(t) = \Sigma_{k=1}^n c_k e^{i \lambda_k t}$ be an exponential polynomial.

In the paper "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type" http://www.math.msu.edu/~fedja/Published/paper.ps Nazarov proves an estimate on the maximum value attained by the polynomial $p$ in an interval $I$, in terms of the maximum of $p$ in a subset $E \subset I$.

To be precise he obtains the following estimate:-

$$ \sup_{t \in I} |p(t)| \leq ( \frac{A \mu(I)}{\mu(E)} )^{n-1} \sup_{t\in E} |p(t)|.$$

At one point he mentions that the result holds true for more general functions of the type $$p(t) = \Sigma_{k=1}^n q_k(t) e^{i \lambda_k t}$$ where $q_k(t)$ are algebraic polynomials of degree d_k$; by an "obvious" approximation argument.

It is not clear to me, what exactly is the argument he is suggesting ?

One of the method he uses to obtain an estimate as mentioned above is by using Turan's Lemma. Although in that case one gets exponent $2 n^2$ instead of $n-1$.

$\underline {Turan's Lemma}$ Let $ z_1,\dots,z_n$ be complex numbers, $|z_j|\geq 1, j=1,\dots,n.$ Let $ b_1,\dots, b_n \in \mathbb C $ and $$S_j:= \Sigma_{k=1}^n b_k z_k^j$$ Then $$|S_0| \leq \{\frac{4 e (m+n-1)}{n}\}^{n-1} \max_{j=m+1}^{m+n} |S_j|.$$ As a simple consequence of this result when the value of an exponential polynomial (with constant coefficient) is known for $n$ consecutive term of an arithmetic progression, then one can get an estimate of the value of the polynomial along that arithmetic progression. i.e.,

Let $p(t)=\Sigma_{k=1}^n c_k e^{i \lambda_k t}$ and assume that the value of the polynomial $p(t)$ is known for $t_j=t_0+j \delta$ for $ j= m+1,...,m+n$. Then substitute $b_k=c_k e^{i \lambda_j t_0}$ and $z_k= e^{i \lambda_k \delta}$ and apply Turan's lemma.

The result now follows by Lebesgue's density theorem and some averaging argument.

We might want to get a similar result like Turan's Lemma for the more general type of exponential polynomial $p(t) = \Sigma_{k=1}^n q_k(t) e^{i \lambda_k t}$ where $q_k(t)$ are algebraic polynomials of degree d_k$.

But I doubt this is what he is suggesting here, as later in order to get the sharper result (the one with exponent $n-1$) he uses some weak type estimates it seems. (I have not read this part of the proof yet).

So, what exactly is the obvious approximation argument he is trying to suggest here?