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

Let the $A$-cusps of a dense subset ${\mathcal{P}}^{\ast }\in [\sqrt{N},N]$ of primes be points $\alpha \in \mathbb{R}\mathrm{/}\mathbb{Z}$ that are such that $\mathrm{\mid }{\sum }_{\begin{array}{c}p\in {\mathcal{P}}^{\ast }\end{array}}e(\alpha p)\mathrm{\mid }\ge \mathrm{\mid }{\mathcal{P}}^{\ast }\mathrm{\mid }\mathrm{/}A$. We establish that any $(1\mathrm{/}N)$-well spaced subset of $A$-cusps contains at most $20A^{2}K\log (2A)$ points, where $K=N\mathrm{/}(\mathrm{\mid }{\mathcal{P}}^{\ast }\mathrm{\mid }\log N)$. 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 $\xi +(a\mathrm{/}q)$. We conclude this study by showing that, given $A\ge 2$, the characteristic function $1_{{\mathcal{P}}^{\ast }}$ may be decomposed in the form $1_{{\mathcal{P}}^{\ast }}=(V(z_0)\log N{)}^{-1}{f}^{\mathrm{♭}}+f^{\mathrm{♯}}$ where the trigonometric polynomial of $f^{\mathrm{♯}}$ takes only values $\le \mathrm{\mid }{\mathcal{P}}^{\ast }\mathrm{\mid }\mathrm{/}A$, and~$f^{\mathrm{♭}}$ 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 $V(z_0)={\prod }_{p<z_0}(1-1\mathrm{/}p)$. The function $f^{\mathrm{♭}}$ 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~${\mathcal{P}}^{\ast }$ to problems on integers coprime to the modulus~$M$.