Restriction estimates with sifted integers

TL;DR

This work extends restriction estimates to sifted sequences $\mathcal S(N)$ carved out by residue-classes modulo primes. By developing an enveloping sieve tailored to the sifted set and a dual large-sieve framework, the authors prove $L^{\ell}$ restriction bounds for $\ell>2$ with explicit constants and a smooth transition to the endpoint $\ell=2$. The approach mirrors Green–Tao and relies on a quantitative enveloping-sieve apparatus, a robust large-sieve inequality, and a careful analysis of the density $V(z)$. They also establish Hardy–Littlewood majorant properties under natural density hypotheses and showcase several nontrivial applications, including polynomial-value patterns and sums over structured residue-classes. The results provide a flexible toolkit for restriction phenomena on sifted arithmetic sets with explicit constants dependent on the sieve parameters.

Abstract

Let $\mathcal{P}$ be a subset of primes and for each prime $p\in \mathcal{P}$, consider a subset $\mathcal{L}_p$ of $\mathbb{Z}/p\mathbb{Z}$. We provide restriction estimates with integers $\leq N$ sifted by $(\mathcal{L}_p)_{\substack{p\leq z\\ p\in \mathcal{P}}}$. This generalizes a result of Green-Tao [3] on the restriction estimates.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof