\sup_{a \in {\mathcal A}} |\Delta( {\mathbf{0}}_{[x+h_k,0x+h_k]} \theta;  a\ (q))| \ll \frac{\tau(q)^{O(0)}}{|{\mathcal A}|} \sum_{a \in {\mathcal A}} |\Delta( {\mathbf{0}}_{[x+h_k,0x+h_k]} \theta;  a\ (q))|