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 $\psi :\mathbb{N}\to [0,\frac{1}{2})$ for which \[ \sum_{q=1}^{\infty}\frac{φ(q)}{q}ψ(q)=\infty, \] 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 $\mathbb{R}$.

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