Rudin Inequality, Chang Theorem, primes and squares

2025/01/09 by Olivier Ramaré, Ramaré, Olivier
Mathematics · #11B30 #11N36 #42A05 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2501.05056

openalex publication_date 2025/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

We prove that the set of large values of the trigonometric polynomial over a subset of density of the primes has some additive structure, similarly to what happens for subsets of densities in ℤ/Nℤ but in a weaker form. To do so, we prove large sieve inequalities for dissociate sets X of circle points and functions f whose support~S is finite and respectively in an interval, in the set of primes or in the set of squares. Set T(f,x)=∑_nf(n)exp(2iπnx). These inequalities are of the shape ∑_x\inX|T(f,x)|²≪ |S|‖f‖₂²log(8R/|S|) where R is respectively N, N/log N and √(N). The implied constants depend on the spacement between sumsets of~X.

Related

• The trigonometric polynomial on sums of two squares, an additive problem and generalisation
• Inverse questions for the large sieve
• Lower bounds on the measure of the support of positive and negative parts of trigonometric polynomials
• Large sieve with sparse sets of moduli for ℤ[i]
• On the large sieve with sparse sets of moduli