Rational points near manifolds and metric Diophantine approximation

Abstract

This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarník type theorems for submanifolds of $\mathbb{R}^n$. These problems have attracted a lot of interest since Kleinbock and Margulis proved a related conjecture of Alan Baker and V. G. Sprindžuk. They have been settled for planar curves but remain open in higher dimensions. In this paper, Khintchine and Jarník type divergence theorems are established for arbitrary analytic nondegenerate manifolds regardless of their dimension. The key to establishing these results is the study of the distribution of rational points near manifolds — a very attractive topic in its own right. Here, for the first time, we obtain sharp lower bounds for the number of rational points near nondegenerate manifolds in dimensions $n>2$ and show that they are ubiquitous (that is uniformly distributed).

  • [Badziahin-Levesley-07:MR2347267] Go to document D. Badziahin and J. Levesley, "A note on simultaneous and multiplicative Diophantine approximation on planar curves," Glasg. Math. J., vol. 49, iss. 2, pp. 367-375, 2007.
    @article {Badziahin-Levesley-07:MR2347267, MRKEY = {2347267},
      AUTHOR = {Badziahin, Dzmitry and Levesley, Jason},
      TITLE = {A note on simultaneous and multiplicative {D}iophantine approximation on planar curves},
      JOURNAL = {Glasg. Math. J.},
      FJOURNAL = {Glasgow Mathematical Journal},
      VOLUME = {49},
      YEAR = {2007},
      NUMBER = {2},
      PAGES = {367--375},
      ISSN = {0017-0895},
      MRCLASS = {11J83 (11J13 11K60)},
      MRNUMBER = {2347267},
      MRREVIEWER = {Henrietta Dickinson},
      DOI = {10.1017/S0017089507003722},
      ZBLNUMBER = {1168.11029},
      }
  • [Baker-1976] Go to document R. C. Baker, "Metric Diophantine approximation on manifolds," J. London Math. Soc., vol. 14, iss. 1, pp. 43-48, 1976.
    @article {Baker-1976, MRKEY = {0419396},
      AUTHOR = {Baker, R. C.},
      TITLE = {Metric {D}iophantine approximation on manifolds},
      JOURNAL = {J. London Math. Soc.},
      FJOURNAL = {Journal of the London Mathematical Society. Second Series},
      VOLUME = {14},
      YEAR = {1976},
      NUMBER = {1},
      PAGES = {43--48},
      ISSN = {0024-6107},
      MRCLASS = {10K15 (10F10)},
      MRNUMBER = {0419396},
      MRREVIEWER = {W. W. Adams},
      ZBLNUMBER = {0338.10058},
      DOI ={10.1112/jlms/s2-14.1.43},
      }
  • [Beresnevich-02:MR1905790] Go to document V. V. Beresnevich, "A Groshev type theorem for convergence on manifolds," Acta Math. Hungar., vol. 94, iss. 1-2, pp. 99-130, 2002.
    @article {Beresnevich-02:MR1905790, MRKEY = {1905790},
      AUTHOR = {Beresnevich, Victor V. },
      TITLE = {A {G}roshev type theorem for convergence on manifolds},
      JOURNAL = {Acta Math. Hungar.},
      FJOURNAL = {Acta Mathematica Hungarica},
      VOLUME = {94},
      YEAR = {2002},
      NUMBER = {1-2},
      PAGES = {99--130},
      ISSN = {0236-5294},
      MRCLASS = {11J83},
      MRNUMBER = {1905790},
      MRREVIEWER = {Henrietta Dickinson},
      DOI = {10.1023/A:1015662722298},
      ZBLNUMBER = {0997.11053},
      }
  • [Beresnevich-Bernik-96:MR1387861] V. V. Beresnevich and V. Bernik, "On a metrical theorem of W. Schmidt," Acta Arith., vol. 75, iss. 3, pp. 219-233, 1996.
    @article {Beresnevich-Bernik-96:MR1387861, MRKEY = {1387861},
      AUTHOR = {Beresnevich, Victor V. and Bernik, V.},
      TITLE = {On a metrical theorem of {W}. {S}chmidt},
      JOURNAL = {Acta Arith.},
      FJOURNAL = {Acta Arithmetica},
      VOLUME = {75},
      YEAR = {1996},
      NUMBER = {3},
      PAGES = {219--233},
      ISSN = {0065-1036},
      CODEN = {AARIA9},
      MRCLASS = {11J83},
      MRNUMBER = {1387861},
      MRREVIEWER = {Eugene M. Matveev},
      ZBLNUMBER = {0858.11042},
      }
  • [Beresnevich-Bernik-Dodson-02:MR2069553] V. V. Beresnevich, V. I. Bernik, and M. M. Dodson, "On the Hausdorff dimension of sets of well-approximable points on nondegenerate curves," Dokl. Nats. Akad. Nauk Belarusi, vol. 46, iss. 6, pp. 18-20, 124, 2002.
    @article {Beresnevich-Bernik-Dodson-02:MR2069553, MRKEY = {2069553},
      AUTHOR = {Beresnevich, Victor V. and Bernik, V. I. and Dodson, M. M.},
      TITLE = {On the {H}ausdorff dimension of sets of well-approximable points on nondegenerate curves},
      JOURNAL = {Dokl. Nats. Akad. Nauk Belarusi},
      FJOURNAL = {Doklady Natsional\cprime noĭAkademii Nauk Belarusi},
      VOLUME = {46},
      YEAR = {2002},
      NUMBER = {6},
      PAGES = {18--20, 124},
      ISSN = {0002-354X},
      MRCLASS = {11J83 (11K55)},
      MRNUMBER = {2069553},
      ZBLNUMBER = {1177.11066},
      }
  • [Beresnevich-Bernik-Kleinbock-Margulis-02:MR1944505] V. V. Beresnevich, V. I. Bernik, D. Y. Kleinbock, and G. A. Margulis, "Metric Diophantine approximation: the Khintchine-Groshev theorem for nondegenerate manifolds," Mosc. Math. J., vol. 2, iss. 2, pp. 203-225, 2002.
    @article {Beresnevich-Bernik-Kleinbock-Margulis-02:MR1944505, MRKEY = {1944505},
      AUTHOR = {Beresnevich, Victor V. and Bernik, V. I. and Kleinbock, D. Y. and Margulis, G. A.},
      TITLE = {Metric {D}iophantine approximation: the {K}hintchine-{G}roshev theorem for nondegenerate manifolds},
      NOTE = {dedicated to Yuri I. Manin on the occasion of his 65th birthday},
      JOURNAL = {Mosc. Math. J.},
      FJOURNAL = {Moscow Mathematical Journal},
      VOLUME = {2},
      YEAR = {2002},
      NUMBER = {2},
      PAGES = {203--225},
      ISSN = {1609-3321},
      MRCLASS = {11J83 (11K60)},
      MRNUMBER = {1944505},
      MRREVIEWER = {Henrietta Dickinson},
      ZBLNUMBER = {1013.11039},
      }
  • [Beresnevich-Dickinson-Velani-06:MR2184760] V. V. Beresnevich, D. Dickinson, and S. Velani, "Measure theoretic laws for lim sup sets," Mem. Amer. Math. Soc., vol. 179, iss. 846, p. x, 2006.
    @article {Beresnevich-Dickinson-Velani-06:MR2184760, MRKEY = {2184760},
      AUTHOR = {Beresnevich, Victor V. and Dickinson, Detta and Velani, Sanju},
      TITLE = {Measure theoretic laws for lim sup sets},
      JOURNAL = {Mem. Amer. Math. Soc.},
      FJOURNAL = {Memoirs of the American Mathematical Society},
      VOLUME = {179},
      YEAR = {2006},
      NUMBER = {846},
      PAGES = {x+91},
      ISSN = {0065-9266},
      CODEN = {MAMCAU},
      MRCLASS = {11J83 (11J13 11K60 28A78 28A80)},
      MRNUMBER = {2184760},
      MRREVIEWER = {Dmitry Y. Kleinbock},
      ZBLNUMBER = {1129.11031},
      }
  • [Beresnevich-Dickinson-Velani-07:MR2373145] Go to document V. V. Beresnevich, D. Dickinson, and S. Velani, "Diophantine approximation on planar curves and the distribution of rational points," Ann. of Math., vol. 166, iss. 2, pp. 367-426, 2007.
    @article {Beresnevich-Dickinson-Velani-07:MR2373145, MRKEY = {2373145},
      AUTHOR = {Beresnevich, Victor V. and Dickinson, Detta and Velani, Sanju},
      TITLE = {Diophantine approximation on planar curves and the distribution of rational points},
      NOTE = {with an Appendix II by R. C. Vaughan},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {166},
      YEAR = {2007},
      NUMBER = {2},
      PAGES = {367--426},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11J83 (28A78)},
      MRNUMBER = {2373145},
      MRREVIEWER = {R. C. Baker},
      DOI = {10.4007/annals.2007.166.367},
      ZBLNUMBER = {1137.11048},
      }
  • [Beresnevich-Vaughan-Velani-08-Inhom] Go to document V. V. Beresnevich, R. C. Vaughan, and S. L. Velani, "Inhomogeneous Diophantine approximation on planar curves," Math. Ann., vol. 349, iss. 4, pp. 929-942, 2011.
    @article {Beresnevich-Vaughan-Velani-08-Inhom, MRKEY = {2777039},
      AUTHOR = {Beresnevich, Victor V. and Vaughan, R. C. and Velani, S. L.},
      TITLE = {Inhomogeneous {D}iophantine approximation on planar curves},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {349},
      YEAR = {2011},
      NUMBER = {4},
      PAGES = {929--942},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {11J83 (11J13 11K60)},
      MRNUMBER = {2777039},
      DOI = {10.1007/s00208-010-0548-9},
      ZBLNUMBER = {1221.11158},
      }
  • [Beresnevich-Velani-07:MR2285737] Go to document V. V. Beresnevich and S. L. Velani, "A note on simultaneous Diophantine approximation on planar curves," Math. Ann., vol. 337, iss. 4, pp. 769-796, 2007.
    @article {Beresnevich-Velani-07:MR2285737, MRKEY = {2285737},
      AUTHOR = {Beresnevich, Victor V. and Velani, Sanju L.},
      TITLE = {A note on simultaneous {D}iophantine approximation on planar curves},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {337},
      YEAR = {2007},
      NUMBER = {4},
      PAGES = {769--796},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {11J13 (11J83)},
      MRNUMBER = {2285737},
      MRREVIEWER = {Henrietta Dickinson},
      DOI = {10.1007/s00208-006-0055-1},
      ZBLNUMBER = {1204.11104},
      }
  • [Beresnevich-Velani-09] Go to document V. V. Beresnevich and S. L. Velani, "Ubiquity and a general logarithm law for geodesics," in Dynamical Systems and Diophantine Approximation, , 2009, vol. 22, pp. 21-36.
    @incollection{Beresnevich-Velani-09,
      author = {Beresnevich, Victor V. and Velani, Sanju L.},
      TITLE = {Ubiquity and a general logarithm law for geodesics},
      BOOKTITLE={Dynamical Systems and Diophantine Approximation},
      SERIES={Séminaires Congrès},
      VOLUME={22},
      YEAR={2009},
      PAGES={21--36},
      URL = {http://www.ams.org/bookstore-getitem/item=SECO-19},
      }
  • [BZ] Go to document V. V. Beresnevich and E. Zorin, "Explicit bounds for rational points near planar curves and metric Diophantine approximation," Adv. Math., vol. 225, iss. 6, pp. 3064-3087, 2010.
    @article {BZ, MRKEY = {2729002},
      AUTHOR = {Beresnevich, Victor V. and Zorin, Evgeniy},
      TITLE = {Explicit bounds for rational points near planar curves and metric {D}iophantine approximation},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {225},
      YEAR = {2010},
      NUMBER = {6},
      PAGES = {3064--3087},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {11J83 (11J13 11K60)},
      MRNUMBER = {2729002},
      DOI = {10.1016/j.aim.2010.05.021},
      ZBLNUMBER = {1207.11076},
      }
  • [Bernik-73:MR0337863] V. I. Bernik, "Asymptotic behavior of the number of solutions of certain systems of inequalities in the theory of Diophantine approximations of dependent variables," Vesc\=\i Akad. Navuk BSSR Ser. F\=\i z.-Mat. Navuk, iss. 1, pp. 10-17, 134, 1973.
    @article {Bernik-73:MR0337863, MRKEY = {0337863},
      AUTHOR = {Bernik, V. I.},
      TITLE = {Asymptotic behavior of the number of solutions of certain systems of inequalities in the theory of {D}iophantine approximations of dependent variables},
      JOURNAL = {Vesc\=\i Akad. Navuk BSSR Ser. F\=\i z.-Mat. Navuk},
      FJOURNAL = {Vesc\=\i Akadèm\=\i \=\i Navuk BSSR. Seryja F\=\i z\=\i ka-Matèmatyčnyh Navuk},
      YEAR = {1973},
      NUMBER = {1},
      PAGES = {10--17, 134},
      ISSN = {0002-3574},
      MRCLASS = {10K15 (10F10)},
      MRNUMBER = {0337863},
      MRREVIEWER = {F. Schweiger},
      ZBLNUMBER={0269.10016},
      }
  • [Bernik-77:MR0480402] V. I. Bernik, "An analogue of Hinčin’s theorem in the metric theory of Diophantine approximations of dependent variables. I," Vesc\=\i Akad. Navuk BSSR Ser. F\=\i z.-Mat. Navuk, iss. 6, pp. 44-49, 141, 1977.
    @article {Bernik-77:MR0480402, MRKEY = {0480402},
      AUTHOR = {Bernik, V. I.},
      TITLE = {An analogue of {H}inčin's theorem in the metric theory of {D}iophantine approximations of dependent variables. {I}},
      JOURNAL = {Vesc\=\i Akad. Navuk BSSR Ser. F\=\i z.-Mat. Navuk},
      FJOURNAL = {Vesc\=\i Akadèm\=\i \=\i Navuk BSSR. Seryja F\=\i z\=\i ka-Matèmatyčnyh Navuk},
      YEAR = {1977},
      NUMBER = {6},
      PAGES = {44--49, 141},
      ISSN = {0002-3574},
      MRCLASS = {10K15},
      MRNUMBER = {0480402},
      MRREVIEWER = {J. Kubilius},
      ZBLNUMBER={0394.10017},
      }
  • [Bernik-1979] V. I. Bernik, "The exact order of approximation of almost all points of a parabola," Mat. Zametki, vol. 26, iss. 5, pp. 657-665, 813, 1979.
    @article {Bernik-1979, MRKEY = {0555016},
      AUTHOR = {Bernik, V. I.},
      TITLE = {The exact order of approximation of almost all points of a parabola},
      JOURNAL = {Mat. Zametki},
      FJOURNAL = {Akademiya Nauk Soyuza SSR. Matematicheskie Zametki},
      VOLUME = {26},
      YEAR = {1979},
      NUMBER = {5},
      PAGES = {657--665, 813},
      ISSN = {0025-567X},
      MRCLASS = {10K15},
      MRNUMBER = {0555016},
      MRREVIEWER = {F. Schweiger},
      ZBLNUMBER = {0415.10047},
      }
  • [Bernik-1983a] V. I. Bernik, "Application of the Hausdorff dimension in the theory of Diophantine approximations," Acta Arith., vol. 42, iss. 3, pp. 219-253, 1983.
    @article {Bernik-1983a, MRKEY = {0729734},
      AUTHOR = {Bernik, V. I.},
      TITLE = {Application of the {H}ausdorff dimension in the theory of {D}iophantine approximations},
      JOURNAL = {Acta Arith.},
      FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
      VOLUME = {42},
      YEAR = {1983},
      NUMBER = {3},
      PAGES = {219--253},
      ISSN = {0065-1036},
      CODEN = {AARIA9},
      MRCLASS = {11K60 (11J81 11K55)},
      MRNUMBER = {0729734},
      MRREVIEWER = {B. Volkmann},
      ZBLNUMBER = {0655.10051},
      ZBLNUMBER = {0482.10049},
      }
  • [Bernik-03:MR2163817] V. I. Bernik, "The Khinchin transference principle and lower bounds for the number of rational points near smooth manifolds," Dokl. Nats. Akad. Nauk Belarusi, vol. 47, iss. 2, pp. 26-28, 2003.
    @article {Bernik-03:MR2163817, MRKEY = {2163817},
      AUTHOR = {Bernik, V. I.},
      TITLE = {The {K}hinchin transference principle and lower bounds for the number of rational points near smooth manifolds},
      JOURNAL = {Dokl. Nats. Akad. Nauk Belarusi},
      FJOURNAL = {Doklady Natsional\cprime noĭAkademii Nauk Belarusi},
      VOLUME = {47},
      YEAR = {2003},
      NUMBER = {2},
      PAGES = {26--28},
      ISSN = {0002-354X},
      MRCLASS = {11J83 (11J61 11K60)},
      MRNUMBER = {2163817},
      MRREVIEWER = {Ela I. Kavaleuskaya},
      ZBLNUMBER = {1204.11116},
      }
  • [BernikDodson-1999] Go to document V. I. Bernik and M. M. Dodson, Metric Diophantine Approximation on Manifolds, Cambridge: Cambridge Univ. Press, 1999, vol. 137.
    @book {BernikDodson-1999, MRKEY = {1727177},
      AUTHOR = {Bernik, V. I. and Dodson, M. M.},
      TITLE = {Metric {D}iophantine Approximation on Manifolds},
      SERIES = {Cambridge Tracts in Math.},
      VOLUME = {137},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1999},
      PAGES = {xii+172},
      ISBN = {0-521-43275-8},
      MRCLASS = {11J83 (11-02 11K55 37A45 37C45)},
      MRNUMBER = {1727177},
      MRREVIEWER = {Dmitry Y. Kleinbock},
      DOI = {10.1017/CBO9780511565991},
      ZBLNUMBER = {0933.11040},
      }
  • [Bernik-Kleinbock-Margulis-01:MR1829381] Go to document V. I. Bernik, D. Kleinbock, and G. A. Margulis, "Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions," Internat. Math. Res. Notices, iss. 9, pp. 453-486, 2001.
    @article {Bernik-Kleinbock-Margulis-01:MR1829381, MRKEY = {1829381},
      AUTHOR = {Bernik, V. I. and Kleinbock, D. and Margulis, G. A.},
      TITLE = {Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions},
      JOURNAL = {Internat. Math. Res. Notices},
      FJOURNAL = {International Mathematics Research Notices},
      YEAR = {2001},
      NUMBER = {9},
      PAGES = {453--486},
      ISSN = {1073-7928},
      MRCLASS = {11J83},
      MRNUMBER = {1829381},
      MRREVIEWER = {Henrietta Dickinson},
      DOI = {10.1155/S1073792801000241},
      ZBLNUMBER = {0986.11053},
      }
  • [Bocher-00:MR1503482] Go to document M. Bôcher, "The theory of linear dependence," Ann. of Math., vol. 2, iss. 1-4, pp. 81-96, 1900/01.
    @article {Bocher-00:MR1503482, MRKEY = {1503482},
      AUTHOR = {B{ô}cher, Maxime},
      TITLE = {The theory of linear dependence},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {2},
      YEAR = {1900/01},
      NUMBER = {1-4},
      PAGES = {81--96},
      ISSN = {0003-486X},
      MRCLASS = {Contributed Item},
      MRNUMBER = {1503482},
      DOI = {10.2307/2007186},
      JFMNUMBER={32.0106.02},
      }
  • [Budarina-Dickinson-07:MR2397137] N. Budarina and D. Dickinson, "Simultaneous Diophantine approximation on surfaces defined by polynomial expressions $x^d_1+\dots+ x^d_m$," in Analytic and Probabilistic Methods in Number Theory/Analiziniai ir Tikimybiniai Metodai Skaiči\polhk u Teorijoje, Vilnius: TEV, 2007, pp. 17-23.
    @incollection {Budarina-Dickinson-07:MR2397137, MRKEY = {2397137},
      AUTHOR = {Budarina, N. and Dickinson, D.},
      TITLE = {Simultaneous {D}iophantine approximation on surfaces defined by polynomial expressions {$x\sp d\sb 1+\dots+ x\sp d\sb m$}},
      BOOKTITLE = {Analytic and Probabilistic Methods in Number Theory/{A}naliziniai ir Tikimybiniai Metodai Skaiči\polhk u Teorijoje},
      PAGES = {17--23},
      PUBLISHER = {TEV},
      ADDRESS = {Vilnius},
      YEAR = {2007},
      MRCLASS = {11J13 (11K55 11K60)},
      MRNUMBER = {2397137},
      MRREVIEWER = {Daniel Berend},
      ZBLNUMBER = {1157.11032},
      }
  • [DickinsonDodson-2000a] Go to document H. Dickinson and M. M. Dodson, "Extremal manifolds and Hausdorff dimension," Duke Math. J., vol. 101, iss. 2, pp. 271-281, 2000.
    @article {DickinsonDodson-2000a, MRKEY = {1738177},
      AUTHOR = {Dickinson, H. and Dodson, M. M.},
      TITLE = {Extremal manifolds and {H}ausdorff dimension},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {101},
      YEAR = {2000},
      NUMBER = {2},
      PAGES = {271--281},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {11J83 (11J20 11K55)},
      MRNUMBER = {1738177},
      MRREVIEWER = {Dmitry Y. Kleinbock},
      DOI = {10.1215/S0012-7094-00-10126-3},
      ZBLNUMBER = {0973.11073},
      }
  • [DickinsonDodson-2001a] Go to document H. Dickinson and M. M. Dodson, "Simultaneous Diophantine approximation on the circle and Hausdorff dimension," Math. Proc. Cambridge Philos. Soc., vol. 130, iss. 3, pp. 515-522, 2001.
    @article {DickinsonDodson-2001a, MRKEY = {1816807},
      AUTHOR = {Dickinson, H. and Dodson, M. M.},
      TITLE = {Simultaneous {D}iophantine approximation on the circle and {H}ausdorff dimension},
      JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
      FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society},
      VOLUME = {130},
      YEAR = {2001},
      NUMBER = {3},
      PAGES = {515--522},
      ISSN = {0305-0041},
      CODEN = {MPCPCO},
      MRCLASS = {11J13 (11J83 28A78)},
      MRNUMBER = {1816807},
      MRREVIEWER = {Dmitry Y. Kleinbock},
      DOI = {10.1017/S0305004101004984},
      ZBLNUMBER = {0992.11046},
      }
  • [DodsonRynneVickers-1989b] Go to document M. M. Dodson, B. P. Rynne, and J. A. G. Vickers, "Metric Diophantine approximation and Hausdorff dimension on manifolds," Math. Proc. Cambridge Philos. Soc., vol. 105, iss. 3, pp. 547-558, 1989.
    @article {DodsonRynneVickers-1989b, MRKEY = {0985691},
      AUTHOR = {Dodson, M. M. and Rynne, B. P. and Vickers, J. A. G.},
      TITLE = {Metric {D}iophantine approximation and {H}ausdorff dimension on manifolds},
      JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
      FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society},
      VOLUME = {105},
      YEAR = {1989},
      NUMBER = {3},
      PAGES = {547--558},
      ISSN = {0305-0041},
      CODEN = {MPCPCO},
      MRCLASS = {11J83 (11K60)},
      MRNUMBER = {0985691},
      MRREVIEWER = {B. Volkmann},
      DOI = {10.1017/S0305004100077926},
      ZBLNUMBER = {0677.10040},
      }
  • [DodsonRynneVickers-1991a] M. M. Dodson, B. P. Rynne, and J. A. G. Vickers, "Khintchine-type theorems on manifolds," Acta Arith., vol. 57, iss. 2, pp. 115-130, 1991.
    @article {DodsonRynneVickers-1991a, MRKEY = {1092764},
      AUTHOR = {Dodson, M. M. and Rynne, B. P. and Vickers, J. A. G.},
      TITLE = {Khintchine-type theorems on manifolds},
      JOURNAL = {Acta Arith.},
      FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
      VOLUME = {57},
      YEAR = {1991},
      NUMBER = {2},
      PAGES = {115--130},
      ISSN = {0065-1036},
      CODEN = {AARIA9},
      MRCLASS = {11J83},
      MRNUMBER = {1092764},
      ZBLNUMBER = {0736.11040},
      }
  • [Dodson-Rynne-Vickers-1996] Go to document M. M. Dodson, B. P. Rynne, and J. A. G. Vickers, "Simultaneous Diophantine approximation and asymptotic formulae on manifolds," J. Number Theory, vol. 58, iss. 2, pp. 298-316, 1996.
    @article {Dodson-Rynne-Vickers-1996, MRKEY = {1393618},
      AUTHOR = {Dodson, M. M. and Rynne, B. P. and Vickers, J. A. G.},
      TITLE = {Simultaneous {D}iophantine approximation and asymptotic formulae on manifolds},
      JOURNAL = {J. Number Theory},
      FJOURNAL = {Journal of Number Theory},
      VOLUME = {58},
      YEAR = {1996},
      NUMBER = {2},
      PAGES = {298--316},
      ISSN = {0022-314X},
      CODEN = {JNUTA9},
      MRCLASS = {11J68},
      MRNUMBER = {1393618},
      MRREVIEWER = {Michael A. Bennett},
      DOI = {10.1006/jnth.1996.0079},
      ZBLNUMBER = {0858.11041},
      }
  • [Drutu-05:MR2195121] Go to document C. Dructu, "Diophantine approximation on rational quadrics," Math. Ann., vol. 333, iss. 2, pp. 405-469, 2005.
    @article {Drutu-05:MR2195121, MRKEY = {2195121},
      AUTHOR = {Dru{\c{t}}u, Cornelia},
      TITLE = {Diophantine approximation on rational quadrics},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {333},
      YEAR = {2005},
      NUMBER = {2},
      PAGES = {405--469},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {11J83 (22E40)},
      MRNUMBER = {2195121},
      MRREVIEWER = {Henrietta Dickinson},
      DOI = {10.1007/s00208-005-0683-x},
      ZBLNUMBER = {1082.11047},
      }
  • [Elkies-2000] Go to document N. D. Elkies, "Rational points near curves and small nonzero $\vert x^3-y^2\vert $ via lattice reduction," in Algorithmic Number Theory, New York: Springer-Verlag, 2000, vol. 1838, pp. 33-63.
    @incollection {Elkies-2000, MRKEY = {1850598},
      AUTHOR = {Elkies, Noam D.},
      TITLE = {Rational points near curves and small nonzero {$\vert x\sp 3-y\sp 2\vert $} via lattice reduction},
      BOOKTITLE = {Algorithmic Number Theory},
      VENUE={{L}eiden, 2000},
      SERIES = {Lecture Notes in Comput. Sci.},
      VOLUME = {1838},
      PAGES = {33--63},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2000},
      MRCLASS = {11D75 (11G50 11H55 11J25)},
      MRNUMBER = {1850598},
      MRREVIEWER = {R. C. Baker},
      DOI = {10.1007/10722028_2},
      ZBLNUMBER = {0991.11072},
      }
  • [GGN] A. Ghosh, A. Gorodnik, and A. Nevo, Diophantine approximation and automorphic spectrum.
    @misc{GGN,
      author={Ghosh, A. and Gorodnik, A. and Nevo, A.},
      TITLE={Diophantine approximation and automorphic spectrum},
      NOTE={preprint},
      ARXIV={1007.0593},
     }
  • [Gorodnik-Shah-08] Go to document A. Gorodnik and N. A. Shah, "Khinchin’s theorem for approximation by integral points on quadratic varieties," Math. Ann., vol. 350, iss. 2, pp. 357-380, 2011.
    @article {Gorodnik-Shah-08, MRKEY = {2794914},
      AUTHOR = {Gorodnik, Alexander and Shah, Nimish A.},
      TITLE = {Khinchin's theorem for approximation by integral points on quadratic varieties},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {350},
      YEAR = {2011},
      NUMBER = {2},
      PAGES = {357--380},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {11J83 (22Exx)},
      MRNUMBER = {2794914},
      DOI = {10.1007/s00208-010-0561-z},
      ZBLNUMBER = {05903693},
      }
  • [Harman-03:MR1979906] Go to document G. Harman, "Simultaneous Diophantine approximation and asymptotic formulae on manifolds," Acta Arith., vol. 108, iss. 4, pp. 379-389, 2003.
    @article {Harman-03:MR1979906, MRKEY = {1979906},
      AUTHOR = {Harman, Glyn},
      TITLE = {Simultaneous {D}iophantine approximation and asymptotic formulae on manifolds},
      JOURNAL = {Acta Arith.},
      FJOURNAL = {Acta Arithmetica},
      VOLUME = {108},
      YEAR = {2003},
      NUMBER = {4},
      PAGES = {379--389},
      ISSN = {0065-1036},
      CODEN = {AARIA9},
      MRCLASS = {11K60 (11J20)},
      MRNUMBER = {1979906},
      MRREVIEWER = {Henrietta Dickinson},
      DOI = {10.4064/aa108-4-7},
      ZBLNUMBER = {1076.11047},
      }
  • [Huxley-1994-rational_points] Go to document M. N. Huxley, "The rational points close to a curve," Ann. Scuola Norm. Sup. Pisa Cl. Sci., vol. 21, iss. 3, pp. 357-375, 1994.
    @article {Huxley-1994-rational_points, MRKEY = {1310631},
      AUTHOR = {Huxley, M. N.},
      TITLE = {The rational points close to a curve},
      JOURNAL = {Ann. Scuola Norm. Sup. Pisa Cl. Sci.},
      FJOURNAL = {Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV},
      VOLUME = {21},
      YEAR = {1994},
      NUMBER = {3},
      PAGES = {357--375},
      ISSN = {0391-173X},
      CODEN = {PSNAAI},
      MRCLASS = {11J25 (11K38)},
      MRNUMBER = {1310631},
      MRREVIEWER = {S. W. Graham},
      URL = {http://www.numdam.org/item?id=ASNSP_1994_4_21_3_357_0},
      ZBLNUMBER = {0827.11046},
      }
  • [Huxley-96:MR1420620] M. N. Huxley, Area, lattice points, and exponential sums, New York: The Clarendon Press Oxford University Press, 1996, vol. 13.
    @book {Huxley-96:MR1420620, MRKEY = {1420620},
      AUTHOR = {Huxley, M. N.},
      TITLE = {Area, lattice points, and exponential sums},
      SERIES = {London Mathematical Society Monographs. New Series},
      VOLUME = {13},
      NOTE = {Oxford Science Publications},
      PUBLISHER = {The Clarendon Press Oxford University Press},
      ADDRESS = {New York},
      YEAR = {1996},
      PAGES = {xii+494},
      ISBN = {0-19-853466-3},
      MRCLASS = {11L07 (11J54 11P21)},
      MRNUMBER = {1420620},
      MRREVIEWER = {R. C. Baker},
      ZBLNUMBER = {0861.11002},
      }
  • [Jarnik-1931] Go to document V. Jarn’ik, "Über die simultanen diophantischen Approximationen," Math. Z., vol. 33, iss. 1, pp. 505-543, 1931.
    @article {Jarnik-1931, MRKEY = {1545226},
      AUTHOR = {Jarn{\'ı}k, Vojtěch},
      TITLE = {Über die simultanen diophantischen {A}pproximationen},
      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {33},
      YEAR = {1931},
      NUMBER = {1},
      PAGES = {505--543},
      ISSN = {0025-5874},
      CODEN = {MAZEAX},
      MRCLASS = {Contributed Item},
      MRNUMBER = {1545226},
      DOI = {10.1007/BF01174368},
      ZBLNUMBER = {0001.32403},
      }
  • [Khintchine-1924] 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 {Khintchine-1924, MRKEY = {1512207},
      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},
      CODEN = {MAANA},
      MRCLASS = {Contributed Item},
      MRNUMBER = {1512207},
      DOI = {10.1007/BF01448437},
      JFMNUMBER={50.0125.01},
      }
  • [Khintchine-1925] Go to document A. Khintchine, "Zwei Bemerkungen zu einer Arbeit des Herrn Perron," Math. Z., vol. 22, iss. 1, pp. 274-284, 1925.
    @article {Khintchine-1925, MRKEY = {1544723},
      AUTHOR = {Khintchine, A.},
      TITLE = {Zwei {B}emerkungen zu einer {A}rbeit des {H}errn {P}erron},
      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {22},
      YEAR = {1925},
      NUMBER = {1},
      PAGES = {274--284},
      ISSN = {0025-5874},
      CODEN = {MAZEAX},
      MRCLASS = {Contributed Item},
      MRNUMBER = {1544723},
      DOI = {10.1007/BF01479606},
      JFMNUMBER={51.0157.02},
      }
  • [Khintchine-1926] Go to document A. Khintchine, "Zur metrischen Theorie der diophantischen Approximationen," Math. Z., vol. 24, iss. 1, pp. 706-714, 1926.
    @article {Khintchine-1926, MRKEY = {1544787},
      AUTHOR = {Khintchine, A.},
      TITLE = {Zur metrischen {T}heorie der diophantischen {A}pproximationen},
      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {24},
      YEAR = {1926},
      NUMBER = {1},
      PAGES = {706--714},
      ISSN = {0025-5874},
      CODEN = {MAZEAX},
      MRCLASS = {Contributed Item},
      MRNUMBER = {1544787},
      DOI = {10.1007/BF01216806},
      JFMNUMBER={52.0183.02},
      }
  • [Kleinbock-03:MR1982150] Go to document D. Kleinbock, "Extremal subspaces and their submanifolds," Geom. Funct. Anal., vol. 13, iss. 2, pp. 437-466, 2003.
    @article {Kleinbock-03:MR1982150, MRKEY = {1982150},
      AUTHOR = {Kleinbock, Dmitry},
      TITLE = {Extremal subspaces and their submanifolds},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {13},
      YEAR = {2003},
      NUMBER = {2},
      PAGES = {437--466},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {11J83},
      MRNUMBER = {1982150},
      MRREVIEWER = {Henrietta Dickinson},
      DOI = {10.1007/s000390300011},
      ZBLNUMBER = {1113.11044},
      }
  • [Kleinbock-04:MR2094125] D. Kleinbock, "Baker-Sprindžuk conjectures for complex analytic manifolds," in Algebraic Groups and Arithmetic, Mumbai: Tata Inst. Fund. Res., 2004, pp. 539-553.
    @incollection {Kleinbock-04:MR2094125, MRKEY = {2094125},
      AUTHOR = {Kleinbock, Dmitry},
      TITLE = {Baker-{S}prindžuk conjectures for complex analytic manifolds},
      BOOKTITLE = {Algebraic Groups and Arithmetic},
      PAGES = {539--553},
      PUBLISHER = {Tata Inst. Fund. Res.},
      ADDRESS = {Mumbai},
      YEAR = {2004},
      MRCLASS = {11J83 (11J54)},
      MRNUMBER = {2094125},
      MRREVIEWER = {Henrietta Dickinson},
      ZBLNUMBER={1103.11020},
      }
  • [Kleinbock-Lindenstrauss-Weiss-04:MR2134453] Go to document D. Kleinbock, E. Lindenstrauss, and B. Weiss, "On fractal measures and Diophantine approximation," Selecta Math., vol. 10, iss. 4, pp. 479-523, 2004.
    @article {Kleinbock-Lindenstrauss-Weiss-04:MR2134453, MRKEY = {2134453},
      AUTHOR = {Kleinbock, Dmitry and Lindenstrauss, Elon and Weiss, Barak},
      TITLE = {On fractal measures and {D}iophantine approximation},
      JOURNAL = {Selecta Math.},
      FJOURNAL = {Selecta Mathematica. New Series},
      VOLUME = {10},
      YEAR = {2004},
      NUMBER = {4},
      PAGES = {479--523},
      ISSN = {1022-1824},
      CODEN = {SMATF6},
      MRCLASS = {11J83},
      MRNUMBER = {2134453},
      MRREVIEWER = {Henrietta Dickinson},
      DOI = {10.1007/s00029-004-0378-2},
      ZBLNUMBER = {1130.11039},
      }
  • [Kleinbock-Margulis-98:MR1652916] Go to document D. Kleinbock and G. A. Margulis, "Flows on homogeneous spaces and Diophantine approximation on manifolds," Ann. of Math., vol. 148, iss. 1, pp. 339-360, 1998.
    @article {Kleinbock-Margulis-98:MR1652916, MRKEY = {1652916},
      AUTHOR = {Kleinbock, Dmitry and Margulis, G. A.},
      TITLE = {Flows on homogeneous spaces and {D}iophantine approximation on manifolds},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {148},
      YEAR = {1998},
      NUMBER = {1},
      PAGES = {339--360},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11J83 (22E40)},
      MRNUMBER = {1652916},
      MRREVIEWER = {Alexander Starkov},
      DOI = {10.2307/120997},
      ZBLNUMBER = {0922.11061},
      }
  • [Kleinbock-Tomanov-07:MR2314053] Go to document D. Kleinbock and G. Tomanov, "Flows on $S$-arithmetic homogeneous spaces and applications to metric Diophantine approximation," Comment. Math. Helv., vol. 82, iss. 3, pp. 519-581, 2007.
    @article {Kleinbock-Tomanov-07:MR2314053, MRKEY = {2314053},
      AUTHOR = {Kleinbock, Dmitry and Tomanov, George},
      TITLE = {Flows on {$S$}-arithmetic homogeneous spaces and applications to metric {D}iophantine approximation},
      JOURNAL = {Comment. Math. Helv.},
      FJOURNAL = {Commentarii Mathematici Helvetici. A Journal of the Swiss Mathematical Society},
      VOLUME = {82},
      YEAR = {2007},
      NUMBER = {3},
      PAGES = {519--581},
      ISSN = {0010-2571},
      MRCLASS = {11J83 (11J54 11J61 37A17 37A45 37D40)},
      MRNUMBER = {2314053},
      MRREVIEWER = {Vladimir S. Anashin},
      DOI = {10.4171/CMH/102},
      ZBLNUMBER = {1135.11037},
      }
  • [Koksma] J. F. Koksma, Diophantische Approximationen, Berlin: Julius Springer, 1936, vol. 4.
    @book{Koksma,
      author={Koksma, J. F.},
      TITLE={Diophantische Approximationen},
      SERIES={Ergebn. Math. Grenzgeb.},
      VOLUME={4},
      YEAR={1936},
      PUBLISHER={Julius Springer},
      ADDRESS={Berlin},
      ZBLNUMBER={0012.39602},
      }
  • [Mahler-1932b] Go to document K. Mahler, "Über das Ma\ss der Menge aller $S$-Zahlen," Math. Ann., vol. 106, iss. 1, pp. 131-139, 1932.
    @article {Mahler-1932b, MRKEY = {1512754},
      AUTHOR = {Mahler, Kurt},
      TITLE = {Über das {M}a\ss der {M}enge aller {$S$}-{Z}ahlen},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {106},
      YEAR = {1932},
      NUMBER = {1},
      PAGES = {131--139},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {Contributed Item},
      MRNUMBER = {1512754},
      DOI = {10.1007/BF01455882},
      JFMNUMBER={0003.24602},
      }
  • [Mashanov-87:MR888593] V. I. Mashanov, "On a problem of Baker in the metric theory of Diophantine approximations," Vests\=\i Akad. Navuk BSSR Ser. F\=\i z.-Mat. Navuk, vol. 1987 , no. 1, pp. 34-38, 125.
    @article {Mashanov-87:MR888593, MRKEY = {0888593},
      AUTHOR = {Mashanov, V. I.},
      TITLE = {On a problem of {B}aker in the metric theory of {D}iophantine approximations},
      JOURNAL = {Vests\=\i Akad. Navuk BSSR Ser. F\=\i z.-Mat. Navuk},
      FJOURNAL = {Vests\=\i Akadèm\=\i \=\i Navuk BSSR. Seryya F\=\i z\=\i ka-Matèmatychnykh Navuk},
      VOLUME = {1987{\rm , no. 1}},
      PAGES = {34--38, 125},
      ISSN = {0002-3574},
      MRCLASS = {11J83 (11J13)},
      MRNUMBER = {0888593},
      MRREVIEWER = {John H. Loxton},
      ZBLNUMBER={0628.11054},
      }
  • [Mazur-04:MR2058289] Go to document B. Mazur, "Perturbations, deformations, and variations (and “near-misses”) in geometry, physics, and number theory," Bull. Amer. Math. Soc., vol. 41, iss. 3, pp. 307-336, 2004.
    @article {Mazur-04:MR2058289, MRKEY = {2058289},
      AUTHOR = {Mazur, B.},
      TITLE = {Perturbations, deformations, and variations (and ``near-misses'') in geometry, physics, and number theory},
      JOURNAL = {Bull. Amer. Math. Soc.},
      FJOURNAL = {American Mathematical Society. Bulletin. New Series},
      VOLUME = {41},
      YEAR = {2004},
      NUMBER = {3},
      PAGES = {307--336},
      ISSN = {0273-0979},
      CODEN = {BAMOAD},
      MRCLASS = {11-02 (11F23 11G18 11J68 14G05 53D55 58K40)},
      MRNUMBER = {2058289},
      MRREVIEWER = {I. Dolgachev},
      DOI = {10.1090/S0273-0979-04-01024-9},
      ZBLNUMBER={1057.11033},
      }
  • [Melnichuk-1979a] Go to document J. V. Melcprimenivcuk, "Diophantine approximations on the circle and Hausdorff dimension," Mat. Zametki, vol. 26, iss. 3, pp. 347-354, 492, 1979.
    @article {Melnichuk-1979a, MRKEY = {0549277},
      AUTHOR = {Mel{\cprime}ni{\v{c}}uk, Ju. V.},
      TITLE = {Diophantine approximations on the circle and {H}ausdorff dimension},
      JOURNAL = {Mat. Zametki},
      FJOURNAL = {Akademiya Nauk Soyuza SSR. Matematicheskie Zametki},
      VOLUME = {26},
      YEAR = {1979},
      NUMBER = {3},
      PAGES = {347--354, 492},
      ISSN = {0025-567X},
      MRCLASS = {10K15 (10K50)},
      MRNUMBER = {0549277},
      MRREVIEWER = {F. Schweiger},
      ZBLNUMBER={0432.10027},
      DOI={10.1007/978-3-540-38645-2},
      }
  • [Pyartli-1969] A. S. Pjartli, "Diophantine approximations of submanifolds of a Euclidean space," Funkcional. Anal. i Priložen., vol. 3, iss. 4, pp. 59-62, 1969.
    @article {Pyartli-1969, MRKEY = {0250978},
      AUTHOR = {Pjartli, A. S.},
      TITLE = {Diophantine approximations of submanifolds of a {E}uclidean space},
      JOURNAL = {Funkcional. Anal. i Priložen.},
      FJOURNAL = {Akademija Nauk SSSR. Funkcional\cprime nyi Analiz i ego Priloženija},
      VOLUME = {3},
      YEAR = {1969},
      NUMBER = {4},
      PAGES = {59--62},
      ISSN = {0374-1990},
      MRCLASS = {10.30 (14.00)},
      MRNUMBER = {0250978},
      MRREVIEWER = {J. B. Roberts},
      }
  • [Schmidt-64:MR0171753] W. M. Schmidt, "Metrische Sätze über simultane Approximation abhängiger Grössen," Monatsh. Math., vol. 68, pp. 154-166, 1964.
    @article {Schmidt-64:MR0171753, MRKEY = {0171753},
      AUTHOR = {Schmidt, Wolfgang M.},
      TITLE = {Metrische {S}ätze über simultane {A}pproximation abhängiger {G}rössen},
      JOURNAL = {Monatsh. Math.},
      VOLUME = {68},
      YEAR = {1964},
      PAGES = {154--166},
      MRCLASS = {10.32},
      MRNUMBER = {0171753},
      MRREVIEWER = {B. Volkmann},
      ZBLNUMBER={0119.28105},
      }
  • [Schmidt-1980] W. M. Schmidt, Diophantine Approximation, New York: Springer-Verlag, 1980, vol. 785.
    @book {Schmidt-1980, MRKEY = {0568710},
      AUTHOR = {Schmidt, Wolfgang M.},
      TITLE = {Diophantine Approximation},
      SERIES = {Lecture Notes in Math.},
      VOLUME = {785},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1980},
      PAGES = {x+299},
      ISBN = {3-540-09762-7},
      MRCLASS = {10Fxx (10-02)},
      MRNUMBER = {0568710},
      MRREVIEWER = {A. J. van der Poorten},
      ZBLNUMBER = {0421.10019},
      }
  • [Sprindzuk-1969-Mahler-problem] V. G. Sprindvzuk, Mahler’s Problem in Metric Number Theory, Providence, R.I.: Amer. Math. Soc., 1969, vol. 25.
    @book {Sprindzuk-1969-Mahler-problem, MRKEY = {0245527},
      AUTHOR = {Sprind{ž}uk, V. G.},
      TITLE = {Mahler's Problem in Metric Number Theory},
      SERIES = {Transl. Math. Monogr.},
      VOLUME={25},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, R.I.},
      YEAR = {1969},
      PAGES = {vii+192},
      MRCLASS = {10.32},
      MRNUMBER = {0245527},
      ZBLNUMBER={0482.10047},
      }
  • [Sprindzuk-1979-Metrical-theory] V. G. Sprindvzuk, Metric Theory of Diophantine Approximations, New York: John Wiley & Sons, 1979.
    @book{Sprindzuk-1979-Metrical-theory,
      author={Sprind{ž}uk, V. G.},
      TITLE={Metric Theory of {Diophantine} Approximations},
      SERIES={Scripta Series in Math.},
      ADDRESS={New York} , PUBLISHER={John Wiley \& Sons},
      YEAR={1979},
      ZBLNUMBER={0463.10020},
      MRNUMBER={0548467},
      }
  • [Sprindzuk-1980-Achievements] V. G. Sprindvzuk, "Achievements and problems of the theory of Diophantine approximations," Uspekhi Mat. Nauk, vol. 35, iss. 4(214), pp. 3-68, 248, 1980.
    @article {Sprindzuk-1980-Achievements, MRKEY = {0586190},
      AUTHOR = {Sprind{ž}uk, V. G.},
      TITLE = {Achievements and problems of the theory of {D}iophantine approximations},
      JOURNAL = {Uspekhi Mat. Nauk},
      FJOURNAL = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
      VOLUME = {35},
      YEAR = {1980},
      NUMBER = {4(214)},
      PAGES = {3--68, 248},
      ISSN = {0042-1316},
      MRCLASS = {10Fxx (10Bxx 12-02)},
      MRNUMBER = {0586190},
      MRREVIEWER = {S. V. Kotov},
      ZBLNUMBER={0463.10020},
      }
  • [Vaughan-Velani-2007] Go to document R. C. Vaughan and S. Velani, "Diophantine approximation on planar curves: the convergence theory," Invent. Math., vol. 166, iss. 1, pp. 103-124, 2006.
    @article {Vaughan-Velani-2007, MRKEY = {2242634},
      AUTHOR = {Vaughan, R. C. and Velani, S.},
      TITLE = {Diophantine approximation on planar curves: the convergence theory},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {166},
      YEAR = {2006},
      NUMBER = {1},
      PAGES = {103--124},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11J83 (11J13 11K60)},
      MRNUMBER = {2242634},
      MRREVIEWER = {Henrietta Dickinson},
      DOI = {10.1007/s00222-006-0509-9},
      ZBLNUMBER = {1185.11047},
      }
  • [Weyl-Weyl-38:MR1503422] Go to document H. Weyl and J. Weyl, "Meromorphic curves," Ann. of Math., vol. 39, iss. 3, pp. 516-538, 1938.
    @article {Weyl-Weyl-38:MR1503422, MRKEY = {1503422},
      AUTHOR = {Weyl, Hermann and Weyl, Joachim},
      TITLE = {Meromorphic curves},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {39},
      YEAR = {1938},
      NUMBER = {3},
      PAGES = {516--538},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {Contributed Item},
      MRNUMBER = {1503422},
      DOI = {10.2307/1968632},
      ZBLNUMBER = {0019.17202},
      }
  • [Whitney] H. Whitney, Geometric Integration Theory, Princeton, NJ: Princeton Univ. Press, 1957.
    @book {Whitney, MRKEY = {0087148},
      AUTHOR = {Whitney, Hassler},
      TITLE = {Geometric Integration Theory},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, NJ},
      YEAR = {1957},
      PAGES = {xv+387},
      MRCLASS = {53.0X},
      MRNUMBER = {0087148},
      MRREVIEWER = {H. Samelson},
      ZBLNUMBER = {0083.28204},
      }

Authors

Victor Beresnevich

Department of Mathematics
University of York
Heslington
York YO10 5DD
England