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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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##### References:
 [1] Beresnevich, Victor; Bernik, Vasily; Dodson, Maurice; Velani, Sanju, Classical metric {D}iophantine approximation revisited. Analytic Number Theory, 38-61 (2009) · Zbl 1236.11064 [2] Aistleitner, Christoph, A note on the {D}uffin-{S}chaeffer conjecture with slow divergence, Bull. Lond. Math. Soc.. Bulletin of the London Mathematical Society, 46, 164-168 (2014) · Zbl 1292.11089 [3] Aistleitner, Christoph, Decoupling theorems for the {D}uffin-{S}chaeffer problem (2019) [4] Aistleitner, Christoph; Lachmann, Thomas; Munsch, Marc; Technau, Niclas; Zafeiropoulos, Agamemnon, The {D}uffin-{S}chaeffer conjecture with extra divergence, Adv. Math.. Advances in Mathematics, 356, 106808-11 (2019) · Zbl 1451.11082 [5] Beresnevich, Victor; Harman, Glyn; Haynes, Alan; Velani, Sanju, The {D}uffin-{S}chaeffer conjecture with extra divergence {II}, Math. Z.. Mathematische Zeitschrift, 275, 127-133 (2013) · Zbl 1333.11068 [6] Beresnevich, Victor; Velani, Sanju, A mass transference principle and the {D}uffin-{S}chaeffer conjecture for {H}ausdorff measures, Ann. of Math. (2). Annals of Mathematics. Second Series, 164, 971-992 (2006) · Zbl 1148.11033 [7] Catlin, Paul A., Two problems in metric {D}iophantine approximation. {I}, J. Number Theory. Journal of Number Theory, 8, 282-288 (1976) · Zbl 0337.10038 [8] Duffin, R. J.; Schaeffer, A. C., Khintchine’s problem in metric {D}iophantine approximation, Duke Math. J.. Duke Mathematical Journal, 8, 243-255 (1941) · JFM 67.0145.03 [9] Dyson, F. J., A theorem on the densities of sets of integers, J. London Math. Soc.. The Journal of the London Mathematical Society, 20, 8-14 (1945) · Zbl 0061.07408 [10] Erd{\H{o}}s, P., On the distribution of the convergents of almost all real numbers, J. Number Theory. Journal of Number Theory, 2, 425-441 (70) · Zbl 0205.34902 [11] Erd{\H{o}}s, P.; Ko, Chao; Rado, R., Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. (2). The Quarterly Journal of Mathematics. Oxford. Second Series, 12, 313-320 (1961) · Zbl 0100.01902 [12] Gallagher, Patrick, Approximation by reduced fractions, J. Math. Soc. Japan. Journal of the Mathematical Society of Japan, 13, 342-345 (1961) · Zbl 0106.04106 [13] Harman, Glyn, Metric Number Theory, London Math. Soc. Monogr. New Series, 18, xviii+297 pp. (1998) · Zbl 1081.11057 [14] Haynes, Alan K.; Pollington, Andrew D.; Velani, Sanju L., The {D}uffin-{S}chaeffer conjecture with extra divergence, Math. Ann.. Mathematische Annalen, 353, 259-273 (2012) · Zbl 1333.11069 [15] Khintchine, A., Einige {S}\"{a}tze \"{u}ber {K}ettenbr\"{u}che, mit {A}nwendungen auf die {T}heorie der {D}iophantischen {A}pproximationen, Math. Ann.. Mathematische Annalen, 92, 115-125 (1924) · JFM 50.0125.01 [16] Khinchin, A. Ya., Continued Fractions, xii+95 pp. (1997) [17] Koukoulopoulos, Dimitris, The Distribution of Prime Numbers, Grad. Stud. in Math., 203, xii + 356 pp. (2019) · Zbl 1468.11001 [18] Montgomery, Hugh L., Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, CBMS Reg. Conf. Ser. Math., 84, xiv+220 pp. (1994) · Zbl 0814.11001 [19] Pollington, A. D.; Vaughan, R. C., The {$$k$$}-dimensional {D}uffin and {S}chaeffer conjecture, Mathematika. Mathematika. A Journal of Pure and Applied Mathematics, 37, 190-200 (1990) · Zbl 0715.11036 [20] Rosser, J. Barkley; Schoenfeld, Lowell, Approximate formulas for some functions of prime numbers, Illinois J. Math.. Illinois Journal of Mathematics, 6, 64-94 (1962) · Zbl 0122.05001 [21] Roth, Klaus F., Sur quelques ensembles d’entiers, C. R. Acad. Sci. Paris. Comptes Rendus Hebdomadaires des S\'{e}ances de l’Acad\'{e}mie des Sciences, 234, 388-390 (1952) · Zbl 0046.04302 [22] Roth, Klaus F., On certain sets of integers, J. London Math. Soc.. The Journal of the London Mathematical Society, 28, 104-109 (1953) · Zbl 0050.04002 [23] Vaaler, Jeffrey D., On the metric theory of {D}iophantine approximation, Pacific J. Math.. Pacific Journal of Mathematics, 76, 527-539 (1978) · Zbl 0352.10026 [24] Walfisz, A., Ein metrischer {S}atz {\"u}ber {D}iophantische {A}pproximationen, Fundamenta Math., 16, 361-385 (1930) · JFM 56.0896.03
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