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

Abstract

Let $\theta\in\mathbb Q\setminus\{0\}$. We construct an approximating function $\psi:\mathbb N\to[0,\frac12)$ for which \[ \sum_{q=1}^{\infty}\frac{\varphi(q)}{q}\psi(q)=\infty, \] but the set of $x\in[0,1]$ for which \[ |qx-a-\theta|<\psi(q),\qquad \gcd(a,q)=1, \] 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 $\mathbb R$.