Salem Inequality

Question

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 can't 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 applications (and as he says must be well known – maybe under a different name?). I would appreciate some references to such inequalities in general. Also I find it curious that the length of the interval does not seem to depend on $n$ and depends only on $\Delta$.

Answer 1

Following a cue from Wadim, this inequality is Theorem 9.1 in Chapter 5 of Zygmund's Trigonometric series, vol 1. Note that although the book is mostly dealing with trigonometric series, the proof is given for general lacunary $\lambda_k.$ (Salem was a good friend of Zygmund's; see the preface to the book.)