Notes on restriction theory in the primes

We study the mean \(\sum\nolimits_{x \in {\cal X}} {|\sum\nolimits_{p \le N} {{u_p}e(xp){|^\ell}}} \) when ℓ covers the full range [2, ∞) and \({\cal X} \subset \mathbb{R}/\mathbb{Z}\) is a well-spaced set, providing a smooth transition from the case ℓ = 2 to the case ℓ > 2 and improving on the results of J. Bourgain and of B. Green and T. Tao. A uniform Hardy–Littlewood property for the set of primes is established as well as a sharp upper bound for \(\sum\nolimits_{x \in {\cal X}} {|\sum\nolimits_{p \le N} {{u_p}e(xp){|^\ell}}}\) when \({\cal X}\) is small. These results are extended to primes in any interval in a last section, provided the primes are numerous enough therein.