The weak inhomogeneous Duffin-Schaeffer conjecture

Theorem 1.1: for every fixed $\gamma\in\mathbb{R}$ and finite-valued $\psi:\mathbb{N}\to[0,\infty)$ with $\sum_q\phi(q)\psi(q)/q=\infty$, almost every $x$ satisfies $\|qx-\gamma\|<\psi(q)$ for infinitely many $q$; the null set may depend on $\gamma$ and $\psi$. Corollary 1.2 gives the Hausdorff-measure analogue via mass transference. The arithmetic core adapts the Koukoulopoulos-Maynard common-pivot method (as simplified by Green-Walker). It proves no convergence converse, no quantitative asymptotic, no coprime-numerator statement, and no result uniform in the shift.

Theorem 1.1: for every fixed $\gamma\in\mathbb{R}$ and finite-valued $\psi:\mathbb{N}\to[0,\infty)$ with $\sum_q\phi(q)\psi(q)/q=\infty$, almost every $x$ satisfies $\|qx-\gamma\|<\psi(q)$ for infinitely many $q$; the null set may depend on $\gamma$ and $\psi$. Corollary 1.2 gives the Hausdorff-measure analogue via mass transference. The arithmetic core adapts the Koukoulopoulos-Maynard common-pivot method (as simplified by Green-Walker). It proves no convergence converse, no quantitative asymptotic, no coprime-numerator statement, and no result uniform in the shift.