# Salem Inequality

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 $\Delta$.

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In Montgomery's book "Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis" the analogue for the maximum norm is stated on page 89. – Helge Jun 15 '10 at 18:41
The result you just mentioned is precisely what Nazarov uses to obtain the analogus result to arbitrary measurable sets (he calls it Turan Lemma). Its at this point he mentions Salem Inequality : I quote "Often it is desirable to have an upper estimate of the sum of absolute values of coefficients $\Sigma_{k=1}^n |c_k|$ rather than of the maximum $\max_{t\in I} |p(t)|$.... a desired estimate can be derived by using the well known Salem Inequality. – Vagabond Jun 15 '10 at 19:06
I think the prove is just squaring and computing the integral, and estimating the remaining terms by $\frac{|c_k| |c_l|}{\Delta |l -k|}$. Then one applies Cauchy-Schwarz twice to get the inequality. But I might have made a mistake on my scrap paper. – Helge Jun 15 '10 at 21:40
I strongly recommend that you modify the title of the question --- it's preferable if the title is a 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
@Vagabond, Salem is famous by his contributions in the theory of Fourier series (www-history.mcs.st-andrews.ac.uk/Biographies/Salem.html), so the inequality is most probably a spacial one of them. Then it should be reflected in any comprehensive treatise on Fourier series (e.g., Zygmund's). – Wadim Zudilin Jun 15 '10 at 23:43

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.)