On the Duffin-Schaeffer conjecture

Abstract

Let $\psi :\mathbb {N}\to \mathbb {R}_{\geqslant 0}$ be an arbitrary function from the positive integers to the non-negative reals. Consider the set $\mathcal {A}$ of real numbers $\alpha $ for which there are infinitely many reduced fractions $a/q$ such that $|\alpha -a/q|\leqslant \psi (q)/q$. If $\sum _{q=1}^\infty \psi (q)\varphi (q)/q=\infty $, we show that $\mathcal {A}$ has full Lebesgue measure. This answers a question of Duffin and Schaeffer. As a corollary, we also establish a conjecture due to Catlin regarding non-reduced solutions to the inequality $|\alpha – a/q|\leqslant \psi (q)/q$, giving a refinement of Khinchin’s Theorem.

  • [DS-survey-article] V. Beresnevich, V. Bernik, M. Dodson, and S. Velani, "Classical metric Diophantine approximation revisited," in Analytic Number Theory, Cambridge Univ. Press, Cambridge, 2009, pp. 38-61.
    @INCOLLECTION{DS-survey-article,
      author = {Beresnevich, Victor and Bernik, Vasily and Dodson, Maurice and Velani, Sanju},
      title = {Classical metric {D}iophantine approximation revisited},
      booktitle = {Analytic Number Theory},
      pages = {38--61},
      publisher = {Cambridge Univ. Press, Cambridge},
      year = {2009},
      mrclass = {11J83 (11K60)},
      mrnumber = {2508636},
      mrreviewer = {Anish Ghosh},
      zblnumber = {1236.11064},
      }
  • [aistleiner] Go to document C. Aistleitner, "A note on the Duffin-Schaeffer conjecture with slow divergence," Bull. Lond. Math. Soc., vol. 46, iss. 1, pp. 164-168, 2014.
    @ARTICLE{aistleiner,
      author = {Aistleitner, Christoph},
      title = {A note on the {D}uffin-{S}chaeffer conjecture with slow divergence},
      journal = {Bull. Lond. Math. Soc.},
      fjournal = {Bulletin of the London Mathematical Society},
      volume = {46},
      year = {2014},
      number = {1},
      pages = {164--168},
      issn = {0024-6093},
      mrclass = {11K60 (11J83)},
      mrnumber = {3784747},
      mrreviewer = {Tomas Persson},
      doi = {10.1112/blms/bdt085},
      url = {https://doi.org/10.1112/blms/bdt085},
      zblnumber = {1292.11089},
      }
  • [aistleitner2] C. Aistleitner, Decoupling theorems for the Duffin-Schaeffer problem, 2019.
    @MISC{aistleitner2,
      author = {Aistleitner, Christoph},
      title = {Decoupling theorems for the {D}uffin-{S}chaeffer problem},
      note = {Progress report, 24pp.},
      year = {2019},
      arxiv = {1907.04590},
      zblnumber = {},
      }
  • [aistleitner] Go to document C. Aistleitner, T. Lachmann, M. Munsch, N. Technau, and A. Zafeiropoulos, "The Duffin-Schaeffer conjecture with extra divergence," Adv. Math., vol. 356, p. 106808, 2019.
    @ARTICLE{aistleitner,
      author = {Aistleitner, Christoph and Lachmann, Thomas and Munsch, Marc and Technau, Niclas and Zafeiropoulos, Agamemnon},
      title = {The {D}uffin-{S}chaeffer conjecture with extra divergence},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {356},
      year = {2019},
      pages = {106808, 11},
      issn = {0001-8708},
      mrclass = {11J83 (11K55 11K60 11N36)},
      mrnumber = {4008524},
      mrreviewer = {Sam Chow},
      doi = {10.1016/j.aim.2019.106808},
      url = {https://doi.org/10.1016/j.aim.2019.106808},
      zblnumber = {07115928},
      }
  • [extra-div2] Go to document V. Beresnevich, G. Harman, A. Haynes, and S. Velani, "The Duffin-Schaeffer conjecture with extra divergence II," Math. Z., vol. 275, iss. 1-2, pp. 127-133, 2013.
    @ARTICLE{extra-div2,
      author = {Beresnevich, Victor and Harman, Glyn and Haynes, Alan and Velani, Sanju},
      title = {The {D}uffin-{S}chaeffer conjecture with extra divergence {II}},
      journal = {Math. Z.},
      fjournal = {Mathematische Zeitschrift},
      volume = {275},
      year = {2013},
      number = {1-2},
      pages = {127--133},
      issn = {0025-5874},
      mrclass = {11J83 (11K55 11K60)},
      mrnumber = {3101800},
      mrreviewer = {Simon Kristensen},
      doi = {10.1007/s00209-012-1126-5},
      url = {https://doi.org/10.1007/s00209-012-1126-5},
      zblnumber = {1333.11068},
      }
  • [hausdorffDS] Go to document V. Beresnevich and S. Velani, "A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures," Ann. of Math. (2), vol. 164, iss. 3, pp. 971-992, 2006.
    @ARTICLE{hausdorffDS,
      author = {Beresnevich, Victor and Velani, Sanju},
      title = {A mass transference principle and the {D}uffin-{S}chaeffer conjecture for {H}ausdorff measures},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {164},
      year = {2006},
      number = {3},
      pages = {971--992},
      issn = {0003-486X},
      mrclass = {11K60 (11J13 11J83 28A78)},
      mrnumber = {2259250},
      mrreviewer = {Andrew M. Rockett},
      doi = {10.4007/annals.2006.164.971},
      url = {https://doi.org/10.4007/annals.2006.164.971},
      zblnumber = {1148.11033},
      }
  • [catlin] Go to document P. A. Catlin, "Two problems in metric Diophantine approximation. I," J. Number Theory, vol. 8, iss. 3, pp. 282-288, 1976.
    @ARTICLE{catlin,
      author = {Catlin, Paul A.},
      title = {Two problems in metric {D}iophantine approximation. {I}},
      journal = {J. Number Theory},
      fjournal = {Journal of Number Theory},
      volume = {8},
      year = {1976},
      number = {3},
      pages = {282--288},
      issn = {0022-314X},
      mrclass = {10K15},
      mrnumber = {0417098},
      mrreviewer = {F. Schweiger},
      doi = {10.1016/0022-314X(76)90006-8},
      url = {https://doi.org/10.1016/0022-314X(76)90006-8},
      zblnumber = {0337.10038},
      }
  • [DS] Go to document R. J. Duffin and A. C. Schaeffer, "Khintchine’s problem in metric Diophantine approximation," Duke Math. J., vol. 8, pp. 243-255, 1941.
    @ARTICLE{DS,
      author = {Duffin, R. J. and Schaeffer, A. C.},
      title = {Khintchine's problem in metric {D}iophantine approximation},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {8},
      year = {1941},
      pages = {243--255},
      issn = {0012-7094},
      mrclass = {10.0X},
      mrnumber = {0004859},
      mrreviewer = {D. C. Spencer},
      doi = {10.1215/S0012-7094-41-00818-9},
      url = {https://doi.org/10.1215/S0012-7094-41-00818-9},
      jfmnumber = {67.0145.03},
      }
  • [dyson] Go to document F. J. Dyson, "A theorem on the densities of sets of integers," J. London Math. Soc., vol. 20, pp. 8-14, 1945.
    @ARTICLE{dyson,
      author = {Dyson, F. J.},
      title = {A theorem on the densities of sets of integers},
      journal = {J. London Math. Soc.},
      fjournal = {The Journal of the London Mathematical Society},
      volume = {20},
      year = {1945},
      pages = {8--14},
      issn = {0024-6107},
      mrclass = {10.0X},
      mrnumber = {0015074},
      mrreviewer = {H. B. Mann},
      doi = {10.1112/jlms/s1-20.1.8},
      url = {https://doi.org/10.1112/jlms/s1-20.1.8},
      zblnumber = {0061.07408},
      }
  • [erdos] Go to document P. ErdHos, "On the distribution of the convergents of almost all real numbers," J. Number Theory, vol. 2, pp. 425-441, 70.
    @ARTICLE{erdos,
      author = {Erd{ő}s, P.},
      title = {On the distribution of the convergents of almost all real numbers},
      journal = {J. Number Theory},
      fjournal = {Journal of Number Theory},
      volume = {2},
      year = {70},
      pages = {425--441},
      issn = {0022-314X},
      mrclass = {10.50},
      mrnumber = {0271058},
      mrreviewer = {D. G. Cantor},
      doi = {10.1016/0022-314X(70)90046-6},
      url = {https://doi.org/10.1016/0022-314X(70)90046-6},
      zblnumber = {0205.34902},
      }
  • [erdoskorado] Go to document P. ErdHos, C. Ko, and R. Rado, "Intersection theorems for systems of finite sets," Quart. J. Math. Oxford Ser. (2), vol. 12, pp. 313-320, 1961.
    @ARTICLE{erdoskorado,
      author = {Erd{ő}s, P. and Ko, Chao and Rado, R.},
      title = {Intersection theorems for systems of finite sets},
      journal = {Quart. J. Math. Oxford Ser. (2)},
      fjournal = {The Quarterly Journal of Mathematics. Oxford. Second Series},
      volume = {12},
      year = {1961},
      pages = {313--320},
      issn = {0033-5606},
      mrclass = {04.60},
      mrnumber = {0140419},
      mrreviewer = {S. Ginsburg},
      doi = {10.1093/qmath/12.1.313},
      url = {https://doi.org/10.1093/qmath/12.1.313},
      zblnumber = {0100.01902},
      }
  • [gallagher] Go to document P. Gallagher, "Approximation by reduced fractions," J. Math. Soc. Japan, vol. 13, pp. 342-345, 1961.
    @ARTICLE{gallagher,
      author = {Gallagher, Patrick},
      title = {Approximation by reduced fractions},
      journal = {J. Math. Soc. Japan},
      fjournal = {Journal of the Mathematical Society of Japan},
      volume = {13},
      year = {1961},
      pages = {342--345},
      issn = {0025-5645},
      mrclass = {10.30},
      mrnumber = {0133297},
      mrreviewer = {J. W. S. Cassels},
      doi = {10.2969/jmsj/01340342},
      url = {https://doi.org/10.2969/jmsj/01340342},
      zblnumber = {0106.04106},
      }
  • [harman] G. Harman, Metric Number Theory, The Clarendon Press, Oxford Univ. Press, New York, 1998, vol. 18.
    @BOOK{harman,
      author = {Harman, Glyn},
      title = {Metric Number Theory},
      series = {London Math. Soc. Monogr. New Series},
      volume = {18},
      publisher = {The Clarendon Press, Oxford Univ. Press, New York},
      year = {1998},
      pages = {xviii+297},
      isbn = {0-19-850083-1},
      mrclass = {11J83 (11K38 11K50 11K55)},
      mrnumber = {1672558},
      mrreviewer = {R. C. Baker},
      zblnumber = {1081.11057},
      }
  • [extra-div1] Go to document A. K. Haynes, A. D. Pollington, and S. L. Velani, "The Duffin-Schaeffer conjecture with extra divergence," Math. Ann., vol. 353, iss. 2, pp. 259-273, 2012.
    @ARTICLE{extra-div1,
      author = {Haynes, Alan K. and Pollington, Andrew D. and Velani, Sanju L.},
      title = {The {D}uffin-{S}chaeffer conjecture with extra divergence},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {353},
      year = {2012},
      number = {2},
      pages = {259--273},
      issn = {0025-5831},
      mrclass = {11J83 (11K55 11K60)},
      mrnumber = {2915535},
      mrreviewer = {Simon Kristensen},
      doi = {10.1007/s00208-011-0683-y},
      url = {https://doi.org/10.1007/s00208-011-0683-y},
      zblnumber = {1333.11069},
      }
  • [Khinchin-paper] Go to document A. Khintchine, "Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen," Math. Ann., vol. 92, iss. 1-2, pp. 115-125, 1924.
    @ARTICLE{Khinchin-paper,
      author = {Khintchine, A.},
      title = {Einige {S}ätze über {K}ettenbrüche, mit {A}nwendungen auf die {T}heorie der {D}iophantischen {A}pproximationen},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {92},
      year = {1924},
      number = {1-2},
      pages = {115--125},
      issn = {0025-5831},
      mrclass = {DML},
      mrnumber = {1512207},
      doi = {10.1007/BF01448437},
      url = {https://doi.org/10.1007/BF01448437},
      jfmnumber = {50.0125.01},
      }
  • [Khinchin-book] Y. A. Khinchin, Continued Fractions, Russian ed., Dover Publications, Inc., Mineola, NY, 1997.
    @BOOK{Khinchin-book,
      author = {Khinchin, A. Ya.},
      title = {Continued Fractions},
      edition = {{R}ussian},
      note = {with a preface by B. V. Gnedenko; reprint of the 1964 translation},
      publisher = {Dover Publications, Inc., Mineola, NY},
      year = {1997},
      pages = {xii+95},
      isbn = {0-486-69630-8},
      mrclass = {11A55 (01A75 11-03 11J70)},
      mrnumber = {1451873},
      jfmnumber = {50.0125.01},
      }
  • [dk-book] Go to document D. Koukoulopoulos, The Distribution of Prime Numbers, Amer. Math. Soc., Providence, RI, 2019, vol. 203.
    @BOOK{dk-book,
      author = {Koukoulopoulos, Dimitris},
      title = {The Distribution of Prime Numbers},
      series = {Grad. Stud. in Math.},
      volume = {203},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {2019},
      pages = {xii + 356},
      isbn = {978-1-4704-4754-0},
      mrclass = {11N05 (11-01 11M06 11N35 11N60)},
      mrnumber = {3971232},
      zblnumber = {07134295},
      doi = {10.1090/gsm/203},
      url = {https://doi.org/10.1090/gsm/203},
      }
  • [montgomery] Go to document H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., Providence, RI, 1994, vol. 84.
    @BOOK{montgomery,
      author = {Montgomery, Hugh L.},
      title = {Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis},
      series = {CBMS Reg. Conf. Ser. Math.},
      volume = {84},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {1994},
      pages = {xiv+220},
      isbn = {0-8218-0737-4},
      mrclass = {11-02 (11Kxx 11L07 11Mxx 11Nxx)},
      mrnumber = {1297543},
      mrreviewer = {John B. Friedlander},
      doi = {10.1090/cbms/084},
      url = {https://doi.org/10.1090/cbms/084},
      zblnumber = {0814.11001},
      }
  • [PV] Go to document A. D. Pollington and R. C. Vaughan, "The $k$-dimensional Duffin and Schaeffer conjecture," Mathematika, vol. 37, iss. 2, pp. 190-200, 1990.
    @ARTICLE{PV,
      author = {Pollington, A. D. and Vaughan, R. C.},
      title = {The {$k$}-dimensional {D}uffin and {S}chaeffer conjecture},
      journal = {Mathematika},
      fjournal = {Mathematika. A Journal of Pure and Applied Mathematics},
      volume = {37},
      year = {1990},
      number = {2},
      pages = {190--200},
      issn = {0025-5793},
      mrclass = {11K60 (11J13)},
      mrnumber = {1099767},
      mrreviewer = {P. Bundschuh},
      doi = {10.1112/S0025579300012900},
      url = {https://doi.org/10.1112/S0025579300012900},
      zblnumber = {0715.11036},
      }
  • [RS] Go to document B. J. Rosser and L. Schoenfeld, "Approximate formulas for some functions of prime numbers," Illinois J. Math., vol. 6, pp. 64-94, 1962.
    @ARTICLE{RS,
      author = {Rosser, J. Barkley and Schoenfeld, Lowell},
      title = {Approximate formulas for some functions of prime numbers},
      journal = {Illinois J. Math.},
      fjournal = {Illinois Journal of Mathematics},
      volume = {6},
      year = {1962},
      pages = {64--94},
      issn = {0019-2082},
      mrclass = {10.42},
      mrnumber = {0137689},
      mrreviewer = {B. K. Ghosh},
      doi = {10.1215/ijm/1255631807},
      url = {https://doi.org/10.1215/ijm/1255631807},
      zblnumber = {0122.05001},
      }
  • [roth1] K. F. Roth, "Sur quelques ensembles d’entiers," C. R. Acad. Sci. Paris, vol. 234, pp. 388-390, 1952.
    @ARTICLE{roth1,
      author = {Roth, Klaus F.},
      title = {Sur quelques ensembles d'entiers},
      journal = {C. R. Acad. Sci. Paris},
      fjournal = {Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences},
      volume = {234},
      year = {1952},
      pages = {388--390},
      issn = {0001-4036},
      mrclass = {10.0X},
      mrnumber = {0046374},
      mrreviewer = {P. Erdős},
      zblnumber = {0046.04302},
      }
  • [roth2] Go to document K. F. Roth, "On certain sets of integers," J. London Math. Soc., vol. 28, pp. 104-109, 1953.
    @ARTICLE{roth2,
      author = {Roth, Klaus F.},
      title = {On certain sets of integers},
      journal = {J. London Math. Soc.},
      fjournal = {The Journal of the London Mathematical Society},
      volume = {28},
      year = {1953},
      pages = {104--109},
      issn = {0024-6107},
      mrclass = {10.0X},
      mrnumber = {0051853},
      mrreviewer = {P. Erdős},
      doi = {10.1112/jlms/s1-28.1.104},
      url = {https://doi.org/10.1112/jlms/s1-28.1.104},
      zblnumber = {0050.04002},
      }
  • [vaaler] Go to document J. D. Vaaler, "On the metric theory of Diophantine approximation," Pacific J. Math., vol. 76, iss. 2, pp. 527-539, 1978.
    @ARTICLE{vaaler,
      author = {Vaaler, Jeffrey D.},
      title = {On the metric theory of {D}iophantine approximation},
      journal = {Pacific J. Math.},
      fjournal = {Pacific Journal of Mathematics},
      volume = {76},
      year = {1978},
      number = {2},
      pages = {527--539},
      issn = {0030-8730},
      mrclass = {10K10},
      mrnumber = {0506128},
      mrreviewer = {Gérard Rauzy},
      doi = {10.2140/pjm.1978.76.527},
      url = {https://doi.org/10.2140/pjm.1978.76.527},
      zblnumber = {0352.10026},
      }
  • [walfisz] Go to document A. Walfisz, "Ein metrischer Satz über Diophantische Approximationen," Fundamenta Math., vol. 16, pp. 361-385, 1930.
    @ARTICLE{walfisz,
      author = {Walfisz, A.},
      title = {Ein metrischer {S}atz {ü}ber {D}iophantische {A}pproximationen},
      journal = {Fundamenta Math.},
      volume = {16},
      year = {1930},
      pages = {361--385},
      doi = {10.4064/fm-16-1-361-385},
      url = {https://doi.org/10.4064/fm-16-1-361-385},
      jfmnumber = {56.0896.03},
      }

Authors

Dimitris Koukoulopoulos

Université de Montréal, Montréal, Canada

James Maynard

Mathematical Institute, University of Oxford, Oxford, England