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On the Duffin-Schaeffer conjecture. (English) Zbl 1459.11154
This very well written and important paper settles the long standing Duffin-Schaeffer conjecture along with some related conjectures in metric Diophantine approximation. The conjecture is concerned with the set \[ \Bigg\{ \alpha \in [0,1] : \bigg\vert \alpha - \frac{a}{q}\bigg\vert \le \frac{\psi(q)}{q} \; \text{ for infinitely many coprime } \; p,q \in {\mathbb Z}, q > 0 \Bigg\}, \] where \(\psi: {\mathbb N} \rightarrow [0,\infty)\) is some function. It is an easy consequence of the Borel-Cantelli lemma that if \(\sum_{q=1}^\infty \psi(q) < \infty\), the Lebesgue measure of this set is equal to \(0\). Conversely, it was famously shown by [A. Khintchine, Math. Ann. 92, 115–125 (1924; JFM 50.0125.01)] that if \(q \psi(q)\) is decreasing, the condition that \(\sum_{q=1}^\infty \psi(q) = \infty\) implies that the Lebesgue measure of the set is equal to \(1\), thus providing a complete description under the assumption of monotonicity.
R. J. Duffin and A. C. Schaeffer [Duke Math. J. 8, 243–255 (1941; Zbl 0025.11002)] proved that the monotonicity condition on \(\psi\) is in fact necessary for the validity of Khintchine’s theorem. Letting \(\phi\) denote the Euler totient function, they conjectured that the Lebesgue measure of the set should instead be governed by the series \(\sum_{q=1}^\infty \psi(q)\phi(q)/q\) in the same manner: convergence should imply measure \(0\) and divergence should imply measure \(1\). This long standing conjecture is settled in the affirmative in the present paper.
The proof starts with a series of reductions. Via a mean-and-variance argument, it is shown that a certain second moment bound is sufficient for the conclusion. This bound is subsequently interpreted as a statement on a bipartite graph with a lot of additional arithmetic structure. With arithmetic methods, it is then shown that the existence of a certain highly structured subgraph is sufficient for the conclusion. To conclude, the authors perform a clever iterative procedure on the original graph to deduce the existence of such a subgraph and hence the Duffin-Schaeffer conjecture.
As consequences of the main result, a conjecture of P. A. Catlin [J. Number Theory 8, 282–288, 289–297 (1976; Zbl 0337.10038)], which provides a zero-one law for the corresponding set without the assumption of coprimality, is deduced. Additionally, results on the Hausdorff dimension of the exceptional set when the series is convergent are deduced by appealing to results of V. Beresnevich and S. Velani [Ann. Math. (2) 164, No. 3, 971–992 (2006; Zbl 1148.11033)].

MSC:
11J83 Metric theory
05C40 Connectivity
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