Rudin Inequality, Chang Theorem, primes and squares

TL;DR

This paper investigates the additive structure of large values of trigonometric polynomials over subsets of primes and squares, establishing large sieve inequalities for dissociate sets in these contexts.

Contribution

It introduces new large sieve inequalities for dissociate sets of circle points with applications to primes and squares, extending additive combinatorics tools.

Findings

Large sieve inequalities for dissociate sets in primes and squares

Additive structure of large polynomial values over prime and square subsets

Inequalities depend on the spacing of sumsets of dissociate sets

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~$\mathcal{X}$.