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)?

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 bookClassical and New Inequalities in Analysis, where the starting point of the exposition is the classical Hölder and Minkowski inequalities. $\endgroup$