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arXiv · 2501.05056

Rudin Inequality, Chang Theorem, primes and squares

Olivier Ramaré
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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 $\mathbb{Z}/{N}\mathbb{Z}$ but in a weaker form. To do so, we prove large sieve inequalities for \emph{dissociate sets} $\mathcal{X}$ 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 $T(f,x)=\sum_{n}f(n)\exp(2i\pi nx)$. These inequalities are of the shape $\sum_{x\in\mathcal{X}}|T(f,x)|^2\ll |S|\|f\|_2^2\log(8R/|S|)$ where $R$ is respectively $N$, $N/\log N$ and $\sqrt{N}$. The implied constants depend on the spacement between sumsets of~$\mathcal{X}$.

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Olivier Ramaré. 2025-01-09. Rudin Inequality, Chang Theorem, primes and squares. https://arxiv.org/abs/2501.05056

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