The Weak Inhomogeneous Duffin–Schaeffer Conjecture

We prove the weak inhomogeneous Duffin–Schaeffer conjecture. For every fixed γ ∈ ℝ and finite-valued $\psi :\mathbb{N}\to [0,\infty )$, divergence of ${\sum }_{q\ge 1}\varphi (q)\psi (q)/q$ implies $\Vert qx-\gamma \Vert <\psi (q)$ for infinitely many q, for almost every x. Here ϕ is Euler's totient and $\Vert \cdot \Vert$ is distance to the nearest integer. Numerators need not be reduced, and the shift satisfies no Diophantine restriction.