Restriction estimates with sifted integers

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 $\le N$ sifted by $({\mathcal{L}}_p{)}_{\begin{array}{c}p\le z\\ p\in \mathcal{P}\end{array}}$. This generalizes a result of Green-Tao [3] on the restriction estimates.