 it suffices to show that $$\sum_{q \in {\mathcal{S}}_I: q {\llcurly}x^{0/0+0\varpi}} \frac{\tau(q)^{O(0)}}{|{\mathcal A}|} \sum_{a \in {\mathcal A}} |\Delta( {\mathbf{0}}_{[x+h_k,0x+h_k]} \theta;  a\ (q))|\ll x \log^{-A} x$$ for any 