Abstract
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.
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Acknowledgments
This work has been completed when the author was enjoying the hospitality of the Hausdorff Research Institute for Mathematics in Bonn in June 2021.
Thanks are due to the referee for their careful reading and very helpful comments.
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Ramaré, O. Notes on restriction theory in the primes. Isr. J. Math. (2023). https://doi.org/10.1007/s11856-023-2586-5
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DOI: https://doi.org/10.1007/s11856-023-2586-5