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I'm trying to understand Bourgain's proof of Proposition 1.10 on page 304-307 in On $\Lambda(p)$-subsets of squares which states

Given $p>4$, we have the estimate \begin{align} \left\|\sum_{n=1}^N a_ne^{in^2t} \right\|_{L^p(\mathbb{T})} \lesssim N^{\frac{1}{2}-\frac{2}{p}}\left( \sum^N_{n=1}|a_n|^2\right)^{\frac{1}{2}}. \end{align}

More precisely, I am trying to understand the following on page 305

Let $t_1, \ldots, t_R$ be $1/N^2$-separated points in $[0, 1]$ satisfying \begin{align} \left|\sum^N_1 a_n e^{2\pi i n^2t_r} \right|>\delta N^{\frac{1}{2}} \ \ (1\leq r \leq R). \ \ \ \ \ \ (4.11) \end{align}

Here we already assumed $\sum |a_n|^2 \leq 1$ and $0<\delta<1$. Then the paper continues

Our purpose is to estimate $R$. By linearization, (4.11) yields \begin{align} \sum_{1\leq r, r'\leq R}\left|\sum^N_1 \exp[2\pi i n^2(t_r-t_{r'})] \right|>\delta^2NR^2. \ \ \ \ \ (4.12) \end{align}

Here I have a potentially very elementary question.

  • What does he mean by linearization in this context, that is, how did he deduced (4.12) from (4.11)?
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    $\begingroup$ See: mathoverflow.net/questions/36653/… $\endgroup$
    – Mark Lewko
    May 5 '19 at 16:27
  • $\begingroup$ @MarkLewko Thank you for the very useful link. $\endgroup$ May 5 '19 at 20:34
  • $\begingroup$ \sum_r \eps_r \sum_n a_n e(n^2 t_r) = \sum_n a_n \sum_r \eps_r e(n^2 t_r) \leq (\sum_n \sum_{r, r’} \eps_r \bar{\eps}_{r’} e(n^2 (t_r - t_{r’})))^{1/2} = (\sum_{r, r’} \eps_r \bar{\eps}_{r’} \sum_n e(n^2 (t_r - t_{r’})))^{1/2} \leq (\sum_{r, r’} |\sum_n e(n^2 (t_r - t_{r’}))|)^{1/2}. $\endgroup$
    – alpoge
    Jun 7 '19 at 23:13
  • $\begingroup$ I remember being stuck at this very point while trying to read this same paper some ten years ago. Fortunately I was reading it with a friend, and we figured it out. $\endgroup$ Jun 22 '19 at 20:32
  • $\begingroup$ I think that "linearization" as Bourgain is referring to it is very closely related to what Beckenbach and Bellman call "quasilinearization" in their book Inequalities (p. 23), which is also closely related to the duality of $L^p$ spaces. The idea is to characterize $L^p$ norms as envelopes of linear quantities. For more on quasilinearization as a strategy for proving inequalities, you can look at p. 669 of the book Classical and New Inequalities in Analysis, where the starting point of the exposition is the classical Hölder and Minkowski inequalities. $\endgroup$
    – Alex Ortiz
    Mar 3 at 16:09
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The method given by Mayank Pandey is pretty much it, but here's an alternative presentation:

There exists some choice of signs $c_r\in \mathbb{C}$ with $\lvert c_r\rvert =1$ such that

$$ \delta N^{1/2}R < \sum_{1\leq r\leq R}\lvert \sum_{1\leq n\leq N} a_n e(n^2t_r) \rvert = \sum_{1\leq n\leq N}a_n \sum_{1\leq r\leq R}c_r e(n^2t_r). $$

By the Cauchy-Schwarz inequality

$$ \delta^2NR^2 < \left(\sum \lvert a_n\rvert^2\right)\left(\sum_{1\leq r,r'\leq R}c_rc_{r'}\sum_{1\leq n\leq N}e(n^2(t_r-t_{r'}))\right). $$

The result follows by the triangle inequality and fact that $\sum \lvert a_n\rvert^2\leq 1$. Note that it was not needed in this argument that the points $t_r$ are $1/N^2$-separated.

The 'trick' of changing the order of summation 'through an absolute value sign' (by introducing the sign factors $c_r$ which are then discarded in the final conclusion), following by Cauchy-Schwarz and/or Holder's inequality, is extremely powerful and appears often throughout Bourgain's work (and elsewhere - it also appears under a slightly different guise in the large sieve-type techniques).

It is usually what Bourgain refers to when he invokes linearisation. I'm not sure if the use of this term for this technique appears outside of Bourgain's work -- presumably it refers to being able to create information about linear relations between the $t_r$ from just a pointwise bound.

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  • $\begingroup$ I found the paper Analytic Principle of the Large Sieve by Montgomery where he puts "linearization" as essentially "duality" of the $\ell^p$ spaces. I just thought I would add this reference for people like me who were interested in a definite place to look in the large sieve literature for related content $\endgroup$
    – Alex Ortiz
    Mar 3 at 16:13
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I think it can be shown as follows. Write $$c_r = \overline{\left(\sum_{n\le N} e(n^2t_r)\right)}$$ (where we write $e(t) = e^{2\pi i t}$ for convenience. Then, we obtain that $$\delta^2 NR < \sum_r c_r\sum_n a_n e(n^2t_r) = \sum_{n}a_n\sum_r c_re(n^2t_r)\le\left(\sum_n \bigg|\sum_r c_re(nt_r)\bigg|^2\right)^{1/2} $$ by Cauchy-Schwarz. Then, since we have that $|c_r| > \delta N^{1/2}$, expanding out the square we obtain that $$\delta^4N^2R^2\le \sum_{r, r'} c_r\overline{c_{r'}}\sum_{n} e(n^2(t_r - t_{r'})) \le \delta^2 N\sum_{r, r'}\bigg|\sum_{n} e(n^2(t_r - t_{r'})\bigg| $$ so (4.12) follows by rearranging. I'm not sure if this is the linearisation in the sense Bourgain means.

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