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

Source abstract

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 $$\begin{equation*} \gg \frac{x\log x}{(\log_{2} x)\log_{3} x}. \end{equation*}$$ This improves the classical Erdős--Rankin bound by a factor of $(\log_{2} x)/(\log_{3} x)^{2}$. Although weaker than the strongest known bounds, it shows what the tilt and its associated covering of composite survivors achieve without Maynard sieve weights or a hypergraph covering theorem. The proof uses the prime number theorem, Mertens' reciprocal-prime formula, and elementary probability. The appendices provide a historical survey and self-contained proofs of the classical Erdős--Rankin bound.