Inverse conjecture for the large sieve with additively structured sieving sets

Let $X$ be positive. For each prime $p<X^{1/2}$, let $\Omega_p\subseteq\mathbb{Z}/p\mathbb{Z}$, and let $A\subseteq[1,X]$ be obtained by sieving out the congruence classes in $\Omega_p$. Here a set $\Omega_p$ is additively structured if there is an $a\not\equiv0\pmod p$ such that

$$\left|\sum_{x\in\Omega_p}e\left(\frac{ax}{p}\right)\right|\gg C_pp,$$

where $C_p\gg\log^{-1}p$.

Refined inverse conjecture for the large sieve. Under these assumptions,

$$|A|\ll X^\epsilon.$$

The conjecture predicts that simultaneous additive structure in all the sieving sets forces the sifted set to be very small, refining the usual inverse large-sieve conjecture.