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

Source abstract

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