I have come across this inequality 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 by Nazarov and he calls it by the name of Salem Inequality (which according to him is well known but I cant find a reference).

If I have understood it correctly the Inequality says that if $p$ is an exponential polynomial whose exponents are well separated, then the average value of square of the modulus of $p$ over a sufficiently large interval dominates the sum of the square of the modulus of its coefficients.

Let $p(t) = \Sigma_{k=1}^n c_k e^{ i \lambda_k t}$, where $ \lambda_1<\lambda_2\dots<\lambda_n \in \mathbb R$ and $\lambda_k$'s satisfies a separation condition i.e., $\lambda_{k+1}-\lambda_k \geq \Delta >0$. Let $I$ be an interval of length bigger than $4\pi / \Delta$, then $$\sum_{k=1}^{n} |c_k|^2 \leq \frac{4}{|I|} \int_I |p(t)|^2 dt. $$ How can one prove this Inequality? This surely would have a lot of appliction (and as he says must be well known !! may be by a different name ?). I would appreciate some references to such inequalities in general. Also I find curious that the length of the interval does not seem to depend on $n$ and depends only on $\Delta$.

isa question, and at least it could have more info than "Salem Inequality". Something like:`How do you prove (and what is another name for) the "Salem Inequality"?`

– Theo Johnson-Freyd Jun 15 '10 at 22:30