I proved the following sieve result and - since the proof is quite long and I need to use it in a work - I am looking for a reference to it (or at least something from which it could be proved quickly). Thank you in advance for any suggestion.
Lemma. For each prime number $p$, let $\Omega_p \subsetneq \{0,1,\ldots,p-1\}$ be a set of residues modulo p. Denote by Ω the whole family of the $\Omega_p$'s. Suppose that $|\Omega_p| \leq c$ for all $p$, and that $$\sum_{p \leq x} |\Omega_p|\cdot \frac{\log p}{p} = k \log x + O(1) ,$$ for all $x>1$, where $c,k>0$ are given constants. Then, fixed any $\delta_1, \delta_2 > 0$, we have $$|\{n \leq x : (n \bmod p) \notin \Omega_p, \forall p \in {]y,z]}\}| \ll_{\Omega,\delta_1,\delta_2} x \cdot \left(\frac{\log y}{\log x}\right)^k ,$$ for all $x > 1$, $2 \leq y \leq (\log x)^{\delta_1}$, and $z \geq x^{\delta_2}$.
Note that, assuming wlog $\delta_2 \leq 1/2$, by the large sieve inequality, the result follows immediately from the lower bound $$\sum_{m \leq x^{\delta_2}} h_y(m) \gg_{\Omega,\delta_1,\delta_2} \left(\frac{\log x}{\log y}\right)^k , $$ where $h_y$ is the multiplicative arithmetic function supported on the squarefree integers with prime factors $> y$ and such that $h_y(p) = |\Omega_p| / (p - |\Omega_p|)$, for all $p > y$.
(This question is clearly connected with this previous one, however I preferred to post it separately since I already edited the latter several times.)
