Cusps of primes in dense subsequences -- Bypassing the $W$-trick

Let the $A$-cusps of a dense subset $\mathcal{P}^*\in[\sqrt{N},N]$ of primes be points $\alpha\in\mathbb{R}/\mathbb{Z}$ that are such that $|\sum_{\substack{p\in\mathcal{P}^*}} e(\alpha p)|\ge |\mathcal{P}^*|/A$. We establish that any $(1/N)$-well spaced subset of $A$-cusps contains at most $20A^2K\log(2A)$ points, where $K=N/(|\mathcal{P}^*|\log N)$. We further show that any $B$-cusps~$\xi$ is accompanied, when $B\le \sqrt{A}$, by a large proportion of $A$-cusps of the shape $\xi+(a/q)$. We conclude this study by showing that, given $A\ge2$, the characteristic function $1_{\mathcal{P}^*}$ may be decomposed in the form $1_{\mathcal{P}^*}=(V(z_0)\log N)^{-1}f^\flat +f^\sharp$ where the trigonometric polynomial of $f^\sharp$ takes only values $\le |\mathcal{P}^*|/A$, and~$f^\flat$ is a bounded non-negative function supported on the integers prime to $M$; the parameters $z_0$ and $M$ are given in terms of~$A$, while $V(z_0)=\prod_{p<z_0}(1-1/p)$. The function $f^\flat$ satisfies more regularity properties. In particular, its density with respect to the integers $\le N$ and coprime to~$M$ is again~$K$. This transfers questions on~$\mathcal{P}^*$ to problems on integers coprime to the modulus~$M$.