Notes on restriction theory in the primes

TL;DR

This paper investigates mean value estimates of exponential sums over primes for a wide range of exponents, improving previous bounds and establishing uniform properties related to primes in various sets and intervals.

Contribution

It introduces new bounds for exponential sums over primes for all exponents, extending previous results and establishing uniform Hardy-Littlewood properties for primes.

Findings

Improved bounds for sums over primes for all b4b4; b4b4>2.

Established uniform Hardy-Littlewood property for primes.

Extended results to primes in arbitrary intervals with sufficient density.

Abstract

TO BE PUBLISHED BY ISRAEL JOURNAL OF MATHEMATICS. We study the mean $\sum_{x\in\mathcal{X}} \bigl|\sum_{p\le N}{}u_p e(xp)\bigr|^{\ell}$ 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 $\sum_{x\in\mathcal{X}} \bigl|\sum_{p\le N}{}u_p e(xp)\bigr|^{\ell}$ 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.