Rudin Inequality, Chang Theorem, primes and squares
Olivier Ramaré
TL;DR
This work studies the additive structure of large values of trigonometric polynomials on sparse subsets, notably primes and squares, by developing Rudin-type inequalities within an abstract harmonic-analytic framework. It introduces dissociate sets and a flexible enveloping-sieve toolkit to obtain improved large sieve bounds against prime and square supports, achieving two-fold density savings relative to the full integers. A prime-specific Chang-type theorem emerges, describing the additive structure of large values on primes, while a parallel Selberg-sieve approach yields strong bounds for squares. Together, the results connect harmonic analysis on dissociate sets with sieve methods to illuminate the additive structure of sparse number-theoretic sets and extend Rudin-Chang phenomena beyond dense integer subsets.
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 $T(f,x)=\sum_{n}f(n)\exp(2iπnx)$. 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~$\mathcal{X}$.