Counterexamples to the inhomogeneous Duffin-Schaeffer conjecture for a residual set of shifts

Let $\theta \in \mathbb{Q}\setminus \{0\}$. We construct an approximating function $\psi :\mathbb{N}\to [0,\frac{1}{2})$ for which $$\sum _{q=1}^{\mathrm{\infty }}\frac{\phi (q)}{q}\psi (q)=\mathrm{\infty },$$ but the set of $x\in [0,1]$ for which $$\mathrm{\mid }qx-a-\theta \mathrm{\mid }<\psi (q),\gcd (a,q)=1,$$ holds for infinitely many $(a,q)\in \mathbb{Z}\times \mathbb{N}$ has Lebesgue measure zero. Thus the inhomogeneous analogue of the Duffin–Schaeffer conjecture fails for every nonzero rational shift. By a modification of the construction, we further show that the set of shifts for which the inhomogeneous Duffin–Schaeffer conjecture fails is residual in $\mathbb{R}$.