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Mosc. Math. J., 2002, Volume 2, Number 2, Pages 203–225 (Mi mmj53)  

This article is cited in 43 scientific papers (total in 43 papers)

Metric Diophantine Approximation: The Khintchine–Groshev Theorem for Nondegenerate Manifolds

V. V. Beresnevicha, V. I. Bernika, D. Ya. Kleinbockb, G. A. Margulisc

a Institute of Mathematics, National Academy of Sciences of the Republic of Belarus
b Brandeis University
c Yale University

Abstract: The main objective of this paper is to prove a Khintchine type theorem on divergence of linear Diophantine approximation on nondegenerate manifolds, which completes earlier results for convergence.

Key words and phrases: Diophantine approximation, Khintchine type theorems, metric theory of Diophantine approximation.

DOI: https://doi.org/10.17323/1609-4514-2002-2-2-203-225

Full text: http://www.ams.org/.../abst2-2-2002.html
References: PDF file   HTML file

Bibliographic databases:

MSC: Primary 11J83; Secondary 11K60
Received: January 30, 2002; in revised form April 11, 2002
Language:

Citation: V. V. Beresnevich, V. I. Bernik, D. Ya. Kleinbock, G. A. Margulis, “Metric Diophantine Approximation: The Khintchine–Groshev Theorem for Nondegenerate Manifolds”, Mosc. Math. J., 2:2 (2002), 203–225

Citation in format AMSBIB
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\by V.~V.~Beresnevich, V.~I.~Bernik, D.~Ya.~Kleinbock, G.~A.~Margulis
\paper Metric Diophantine Approximation: The Khintchine--Groshev Theorem for Nondegenerate Manifolds
\jour Mosc. Math.~J.
\yr 2002
\vol 2
\issue 2
\pages 203--225
\mathnet{http://mi.mathnet.ru/mmj53}
\crossref{https://doi.org/10.17323/1609-4514-2002-2-2-203-225}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Kleinbock D., “Extremal subspaces and their submanifolds”, Geom. Funct. Anal., 13:2 (2003), 437–466  crossref  mathscinet  zmath  isi
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    5. Beresnevich V., “On a theorem of V. Bernik in the metric theory of Diophantine approximation”, Acta Arith., 117:1 (2005), 71–80  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. Beresnevich V.V., Bernik V.I., Kovalevskaya E.I., “On approximation of $p$-adic numbers by $p$-adic algebraic numbers”, J. Number Theory, 111:1 (2005), 33–56  crossref  mathscinet  zmath  isi  elib
    7. Beresnevich V., Dickinson D., Velani S., Measure theoretic laws for lim sup sets, Mem. Amer. Math. Soc., 179, no. 846, 2006, x+91 pp.  mathscinet  zmath  isi  elib
    8. Beresnevich V., Dickinson D., Velani S., Vaughan R.C., “Diophantine approximation on planar curves and the distribution of rational points”, Ann. of Math. (2), 166:2 (2007), 367–426  crossref  mathscinet  zmath  isi
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    16. Mohammadi A., Salehi Golsefidy A., “$S$-arithmetic Khintchine-type theorem”, Geom. Funct. Anal., 19:4 (2009), 1147–1170  crossref  mathscinet  zmath  isi  elib
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    18. Budarina N., Dickinson D., “Diophantine approximation on non-degenerate curves with non-monotonic error function”, Bull. Lond. Math. Soc., 41:1 (2009), 137–146  crossref  mathscinet  zmath  isi  elib
    19. Badziahin D., “Inhomogeneous Diophantine approximation on curves and Hausdorff dimension”, Adv. Math., 223:1 (2010), 329–351  crossref  mathscinet  zmath  isi  elib
    20. Beresnevich V., Velani S., “An inhomogeneous transference principle and Diophantine approximation”, Proc London Math Soc, 101:3 (2010), 821–851  crossref  mathscinet  zmath  isi  elib
    21. Beresnevich V., Zorin E., “Explicit bounds for rational points near planar curves and metric Diophantine approximation”, Adv Math, 225:6 (2010), 3064–3087  crossref  mathscinet  zmath  isi  elib
    22. Ghosh A., “Diophantine approximation on affine hyperplanes”, Acta Arith, 144:2 (2010), 167–182  crossref  mathscinet  zmath  isi  elib
    23. Budarina N., Dickinson D., Levesley J., “Simultaneous Diophantine Approximation on Polynomial Curves”, Mathematika, 56:1 (2010), 77–85  crossref  mathscinet  zmath  isi  elib
    24. Budarina N., “Simultaneous Diophantine Approximation in the Real and $p$-Adic Fields with Nonmonotonic Error Function”, Lith Math J, 51:4 (2011), 461–471  crossref  mathscinet  zmath  isi  elib
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    26. Ghosh A., “A Khintchine-Groshev Theorem for Affine Hyperplanes”, Int J Number Theory, 7:4 (2011), 1045–1064  crossref  mathscinet  zmath  isi  elib
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    28. N. V. Budarina, “Metricheskaya teoriya sovmestnykh diofantovykh priblizhenii v $\mathbb{R}^{k}\times\mathbb{C}^{l}\times \mathbb{Q}^m_{p}$”, Chebyshevskii sb., 12:1 (2011), 17–50  mathnet  mathscinet
    29. N. V. Budarina, “Regulyarnye i povsemestnye sistemy dlya sovmestnykh diofantovykh priblizhenii”, Chebyshevskii sb., 12:4 (2011), 43–74  mathnet  mathscinet
    30. Beresnevich V., “Rational points near manifolds and metric Diophantine approximation”, Ann of Math (2), 175:1 (2012), 187–235  crossref  mathscinet  zmath  isi
    31. Mohammadi A., Golsefidy A.S., “Simultaneous Diophantine Approximation in Non-Degenerate P-Adic Manifolds”, Isr. J. Math., 188:1 (2012), 231–258  crossref  mathscinet  zmath  isi  elib
    32. Badziahin D., Beresnevich V., Velani S., “Inhomogeneous Theory of Dual Diophantine Approximation on Manifolds”, Adv. Math., 232:1 (2013), 1–35  crossref  mathscinet  zmath  isi  elib
    33. N. V. Budarina, “Inhomogeneous Diophantine approximation on curves with non-monotonic error function”, Dalnevost. matem. zhurn., 13:2 (2013), 164–178  mathnet
    34. Budarina N., Dickinson D., “Inhomogeneous Diophantine Approximation on Integer Polynomials with Non-Monotonic Error Function”, Acta Arith., 160:3 (2013), 243–257  crossref  mathscinet  zmath  isi  elib
    35. Dodson M., Everitt B., “Metrical Diophantine Approximation For Quaternions”, Math. Proc. Camb. Philos. Soc., 157:3 (2014), 513–542  crossref  mathscinet  zmath  isi  elib
    36. N. V. Budarina, F. Götze, “On regular systems of algebraic $p$-adic numbers of arbitrary degree in small cylinders”, Dalnevost. matem. zhurn., 15:2 (2015), 133–155  mathnet  elib
    37. Hussain M., “a Jarnik Type Theorem For Planar Curves: Everything About the Parabola”, Math. Proc. Camb. Philos. Soc., 159:1 (2015), 47–60  crossref  mathscinet  zmath  isi  elib
    38. Bernik V., Goetze F., “a New Connection Between Metric Theory of Diophantine Approximations and Distribution of Algebraic Numbers”, Recent Trends in Ergodic Theory and Dynamical Systems, Contemporary Mathematics, 631, eds. Bhattacharya S., Das T., Ghosh A., Shah R., Amer Mathematical Soc, 2015, 33–45  crossref  mathscinet  zmath  isi
    39. Beresnevich V., Kleinbock D., Margulis G., “Non-Planarity and Metric Diophantine Approximation For Systems of Linear Forms”, J. Theor. Nr. Bordx., 27:1 (2015), 1–31  crossref  mathscinet  zmath  isi
    40. Badziahin D., Harrap S., Hussain M., “An Inhomogeneous Jarnik Type Theorem For Planar Curves”, Math. Proc. Camb. Philos. Soc., 163:1 (2017), 47–70  crossref  zmath  isi  scopus
    41. V. I. Bernik, A. G. Gusakova, A. S. Kudin, “Otsenki sverkhu i snizu dlya kolichestva algebraicheskikh tochek v korotkikh intervalakh”, Chebyshevskii sb., 18:4 (2017), 116–127  mathnet  crossref
    42. Huang J.-J., “Hausdorff Theory of Dual Approximation on Planar Curves”, J. Reine Angew. Math., 740 (2018), 63–76  crossref  mathscinet  zmath  isi  scopus
    43. Aka M., Breuillard E., Rosenzweig L., de Saxce N., “Diophantine Approximation on Matrices and Lie Groups”, Geom. Funct. Anal., 28:1 (2018), 1–57  crossref  mathscinet  zmath  isi  scopus
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