The weak inhomogeneous Duffin–Schaeffer conjecture

Proves that for every real shift γ and finite-valued $\psi:\mathbb N\to[0,\infty)$, divergence of $\sum_q\phi(q)\psi(q)/q$ implies $\|qx-\gamma\|\lt \psi(q)$ for infinitely many q, for almost every x. Here ϕ is Euler's totient and the norm is distance to the nearest integer. Numerators are unrestricted; no monotonicity or Diophantine condition on γ is needed.