Notes on restriction theory in the primes

2021/09/21 by Ramaré, Olivier · 2 citations
#11N36 #43A46 #FOS: Mathematics #Functional Analysis (math.FA) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2109.10180

Abstract

TO BE PUBLISHED BY ISRAEL JOURNAL OF MATHEMATICS. We study the mean ∑_x\inX |∑_{p≤ N}u_p e(xp)|^ℓ when ℓ covers the full range [2,∞) and X⊂ℝ/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 ∑_x\inX |∑_{p≤ N}u_p e(xp)|^ℓ when 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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• Restriction estimates with sifted integers
• The trigonometric polynomial on sums of two squares, an additive problem and generalisation

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