The Duffin–Schaeffer conjecture with a moving target

The paper proves, for m ≥ 3 and arbitrary moving targets y = (y_q), the zero–full law for W'_m(y,ψ) with the divergence criterion ∑(ϕ(q)ψ(q)/q)^m = ∞, via uniform overlap estimates and a refined Borel–Cantelli criterion (Proposition 2.3) together with careful handling of the large-ψ regime; see the main theorem statement and proof strategy in Theorem 1 and Section 2 . In contrast, the candidate solution assumes a uniform Gallagher-style L^2 quasi-independence bound for the moving-target events without proof and misstates that having infinitely many q with ψ(q) > 1/2 forces ∑(ϕ(q)ψ(q)/q)^m to diverge. The paper explicitly treats the large-ψ case with a nontrivial case split and sieve-theoretic input (e.g., Corollary 2.5 and the estimates leading to (31)), rather than invoking a uniform Gallagher bound . Moreover, the one-dimensional moving-target counterexample (Theorem 2) underscores that naive uniform L^2 arguments cannot be taken for granted in the inhomogeneous moving-target setting . Overall, the paper’s argument is complete and correct, while the model’s outline relies on unproven uniformity and contains a false claim about the large-ψ subsequence.