Notes on restriction theory in the primes

Abstract

TO BE PUBLISHED BY ISRAEL JOURNAL OF MATHEMATICS. We study the mean ${\sum }_{x\in \mathcal{X}}|{\sum }_{p\le N}{u}_{p}e(xp){|}^{\ell }$ when $\ell$ covers the full range $[2,\infty )$ and $\mathcal{X}\subset \mathbb{R}/\mathbb{Z}$ is a {well-spaced} set, providing a smooth transition from the case $\ell =2$ to the case $\ell >2$ 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 ${\sum }_{x\in \mathcal{X}}|{\sum }_{p\le N}{u}_{p}e(xp){|}^{\ell }$ when $\mathcal{X}$ is small. These results are extended to primes in \emph{any} interval in a last section, provided the primes are numerous enough therein.