On the Duffin-Schaeffer conjecture

The authors state in Section 3 that 'the key problem is to win back a little bit more than the x^{o(1)} factor from the divisor bound.' This is a crucial insight, but the transition from the 'Model Problem' to the actual iterative procedure is quite abrupt. It would be beneficial for the authors to provide more intuition on why the specific 'quality' function defined in Definition 6.2 is the correct quantity to track. While the authors mention that 'the ϕ(q)/q weights are vital for our argument to work,' a more explicit discussion of how these weights specifically interact with the 'compression' argument to prevent the failure seen in the counterexample to the Model Problem would significantly improve the reader's understanding of the proof's mechanics.

The dependency structure of the proof, as illustrated in the diagrams in Section 4, is very helpful. However, the sheer number of propositions and lemmas, such as 'Propositions 8.1-8.3 and Lemmas 8.4-8.5,' creates a high barrier to entry. The authors could further assist the reader by providing a brief 'roadmap' paragraph at the beginning of Section 8 that explains the high-level strategy for each of these five components, rather than just listing them. For example, explicitly stating which components handle the 'large prime' case versus the 'small prime' case would help the reader maintain the narrative thread through the technical sections.

The authors state in Section 3 that 'the key to the proof is to show that P(q, r) ≪ 1 on average over q, r ∈ S'. This is a crucial insight, and the subsequent reduction to the 'Model Problem' is well-motivated. However, the transition from the Model Problem to the formal iterative procedure in Section 8 is quite abrupt. It would be beneficial if the authors could provide more explicit commentary on why the 'quality' function defined in Definition 6.6 is the correct quantity to track, as it is not immediately obvious why this specific combination of terms is sufficient to guarantee the convergence of the iteration.

The treatment of the 'small primes' in Section 13 is a masterclass in handling technical obstructions. The authors note that 'the p-adic valuation is exactly k' for the set V_p^k, and the careful management of these sets is what prevents the iteration from collapsing. The use of Lemma 10.1 to ensure that all vertices have high degree is a clever way to maintain the structural integrity of the graph throughout the iteration. I found the explanation of the 'clean-up' step in Section 3 to be particularly helpful in understanding the overall strategy, and I would encourage the authors to ensure that this is clearly linked to the formal definitions in Section 6.

The authors state in Section 3 that 'the ϕ(q)/q weights are vital for our argument to work'. While they provide a brief discussion and a reference to counterexamples in Section 15, it would be beneficial to expand slightly on why the 'Model Problem' fails specifically for the measure μ(v) = ψ(v)ϕ(v)/v compared to a uniform measure. A more explicit connection between the failure of the Model Problem and the specific structure of the weights would deepen the reader's understanding of why this specific measure is the 'natural' one for the conjecture.

In the proof of Proposition 7.1, the authors define the quality q(G) using a product over primes in P, specifically '∏ p ∈P p | f ( p ) − g ( p ) | (1 − 1 f ( p )= g ( p ) > 1 /p ) 2 (1 − 1 /p 31 / 30 ) 10'. The justification for the inclusion of the (1 − 1/p^31/30)^10 factor is described as being 'included merely for convenience'. While this is acceptable, a brief remark on whether the exponent 31/30 is optimal or if any value greater than 1 would suffice would be helpful for specialists who might wish to adapt this method to other problems.

The dependency diagrams provided in Section 4 are an excellent feature and significantly aid in navigating the paper's complex structure. However, the authors note that 'All propositions are proven in sections later than they appear.' This is standard, but the sheer volume of dependencies makes it easy for a reader to lose track of the global strategy. I recommend that the authors add a brief sentence at the start of each major section (e.g., Section 12, 13, 14) explicitly restating the goal of that section in the context of the overall proof of Theorem 1.