A tyro's approach to the tilted sieve: beyond the Erdős--Rankin bound

AI use, as the source states it

“We give an elementary exposition of the tilted sieve introduced by GPT-5.6 Sol \cite{GPT2026}, showing that one residue class modulo each prime p \le x can cover an interval of length \begin{equation*} \gg \frac{x\log x}{(\log_{2} x)\log_{3} x}.”

Model: GPT-5.6 Sol, as stated