Arithmetical trigonometric polynomials at small height rationals

After highlighting some less known features of Vinogradov's initial proof of the three primes Theorem, we will discuss some recent results obtained together with G. Kasi Visvanadham. In particular we will present a family of bilinear forms for the primes or the Moebius function that may be typically used to prove sharp estimates for
$$ \sum\limits_{X\,<\,p\,\leqslant\, X+X^{9/10}}e(pa/q) $$
for any $q\leqslant X^{1/10}$. The method is flexible and adapts to several other cases which we shall discuss if time permits, as well as some applications.