Abstract
For any $i,j \ge 0$ with $i+j =1$, let $\mathbf{Bad}(i,j)$ denote the set of points $(x,y) \in \mathbb{R}^2$ for which $ \max \{ \|qx\|^{1/i}, \, \|qy\|^{1/j} \} > c/q $ for all $ q \in \mathbb{N}$. Here $c = c(x,y)$ is a positive constant. Our main result implies that any finite intersection of such sets has full dimension. This settles a conjecture of Wolfgang M. Schmidt in the theory of simultaneous Diophantine approximation.
-
[cassels] J. W. S. Cassels, An Introduction to the Geometry of Numbers, New York: Springer-Verlag, 1997.
@book {cassels, MRKEY = {1434478},
AUTHOR = {Cassels, J. W. S.},
TITLE = {An Introduction to the Geometry of Numbers},
SERIES = {Classics in Mathematics},
NOTE = {corrected reprint of the 1971 edition},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1997},
PAGES = {viii+344},
ISBN = {3-540-61788-4},
MRCLASS = {11Hxx},
MRNUMBER = {1434478},
ZBLNUMBER = {0866.11041},
} -
[dav]
H. Davenport, "A note on Diophantine approximation. II," Mathematika, vol. 11, pp. 50-58, 1964.@article {dav, MRKEY = {0166154},
AUTHOR = {Davenport, H.},
TITLE = {A note on {D}iophantine approximation. {II}},
JOURNAL = {Mathematika},
FJOURNAL = {Mathematika. A Journal of Pure and Applied Mathematics},
VOLUME = {11},
YEAR = {1964},
PAGES = {50--58},
ISSN = {0025-5793},
MRCLASS = {10.30},
MRNUMBER = {0166154},
MRREVIEWER = {J. Popken},
DOI = {10.1112/S0025579300003478},
ZBLNUMBER = {0122.05903},
} -
[falc] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Chichester: John Wiley & Sons Ltd., 1990.
@book {falc, MRKEY = {1102677},
AUTHOR = {Falconer, Kenneth},
TITLE = {Fractal Geometry: Mathematical Foundations and Applications},
PUBLISHER = {John Wiley \& Sons Ltd.},
ADDRESS = {Chichester},
YEAR = {1990},
PAGES = {xxii+288},
ISBN = {0-471-92287-0},
MRCLASS = {28A80 (00A69 11K55 28-01 58F13 60G18)},
MRNUMBER = {1102677},
MRREVIEWER = {Christoph Bandt},
ZBLNUMBER = {0689.28003},
} -
[KW] D. Kleinbock and B. Weiss, "Modified Schmidt games and Diophantine approximation with weights," Adv. Math., vol. 223, iss. 4, pp. 1276-1298, 2010.
@article {KW, MRKEY = {2581371},
AUTHOR = {Kleinbock, Dmitry and Weiss, Barak},
TITLE = {Modified {S}chmidt games and {D}iophantine approximation with weights},
JOURNAL = {Adv. Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {223},
YEAR = {2010},
NUMBER = {4},
PAGES = {1276--1298},
ISSN = {0001-8708},
CODEN = {ADMTA4},
MRCLASS = {11J83 (11J13)},
MRNUMBER = {2581371},
MRREVIEWER = {Simon Kristensen},
DOI = {10.1016/j.aim.2009.09.018},
ZBLNUMBER = {1213.11148},
} -
[KTV] S. Kristensen, R. Thorn, and S. Velani, "Diophantine approximation and badly approximable sets," Adv. Math., vol. 203, iss. 1, pp. 132-169, 2006.
@article {KTV, MRKEY = {2231044},
AUTHOR = {Kristensen, Simon and Thorn, Rebecca and Velani, Sanju},
TITLE = {Diophantine approximation and badly approximable sets},
JOURNAL = {Adv. Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {203},
YEAR = {2006},
NUMBER = {1},
PAGES = {132--169},
ISSN = {0001-8708},
CODEN = {ADMTA4},
MRCLASS = {11J83 (37C45)},
MRNUMBER = {2231044},
MRREVIEWER = {Dmitry Y. Kleinbock},
DOI = {10.1016/j.aim.2005.04.005},
ZBLNUMBER = {1098.11039},
} -
[pvl] A. D. Pollington and S. L. Velani, "On a problem in simultaneous Diophantine approximation: Littlewood’s conjecture," Acta Math., vol. 185, iss. 2, pp. 287-306, 2000.
@article {pvl, MRKEY = {1819996},
AUTHOR = {Pollington, Andrew D. and Velani, Sanju L.},
TITLE = {On a problem in simultaneous {D}iophantine approximation: {L}ittlewood's conjecture},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {185},
YEAR = {2000},
NUMBER = {2},
PAGES = {287--306},
ISSN = {0001-5962},
CODEN = {ACMAA8},
MRCLASS = {11J25 (11J70 11K60)},
MRNUMBER = {1819996},
MRREVIEWER = {Michel Waldschmidt},
DOI = {10.1007/BF02392812},
ZBLNUMBER = {0970.11026},
} -
[PV] A. D. Pollington and S. L. Velani, "On simultaneously badly approximable numbers," J. London Math. Soc., vol. 66, iss. 1, pp. 29-40, 2002.
@article {PV, MRKEY = {1911218},
AUTHOR = {Pollington, Andrew D. and Velani, Sanju L.},
TITLE = {On simultaneously badly approximable numbers},
JOURNAL = {J. London Math. Soc.},
FJOURNAL = {Journal of the London Mathematical Society. Second Series},
VOLUME = {66},
YEAR = {2002},
NUMBER = {1},
PAGES = {29--40},
ISSN = {0024-6107},
CODEN = {JLMSAK},
MRCLASS = {11J25},
MRNUMBER = {1911218},
MRREVIEWER = {Dmitry Y. Kleinbock},
DOI = {10.1112/S0024610702003265},
ZBLNUMBER = {1026.11061},
} -
[schconj] W. M. Schmidt, "Open problems in Diophantine approximation," in Diophantine Approximations and Transcendental Numbers, Boston, MA: Birkhäuser, 1983, vol. 31, pp. 271-287.
@incollection {schconj, MRKEY = {0702204},
AUTHOR = {Schmidt, W. M.},
TITLE = {Open problems in {D}iophantine approximation},
BOOKTITLE = {Diophantine Approximations and Transcendental Numbers},
VENUE={{L}uminy, 1982},
SERIES = {Progr. Math.},
VOLUME = {31},
PAGES = {271--287},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {1983},
MRCLASS = {11Jxx},
MRNUMBER = {0702204},
MRREVIEWER = {C. G. Lekkerkerker},
ZBLNUMBER = {0529.10032},
} -
[vent] A. Venkatesh, "The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture," Bull. Amer. Math. Soc., vol. 45, iss. 1, pp. 117-134, 2008.
@article {vent, MRKEY = {2358379},
AUTHOR = {Venkatesh, Akshay},
TITLE = {The work of {E}insiedler, {K}atok and {L}indenstrauss on the {L}ittlewood conjecture},
JOURNAL = {Bull. Amer. Math. Soc.},
FJOURNAL = {American Mathematical Society. Bulletin. New Series},
VOLUME = {45},
YEAR = {2008},
NUMBER = {1},
PAGES = {117--134},
ISSN = {0273-0979},
CODEN = {BAMOAD},
MRCLASS = {11J13 (11H46 37A35 37A45)},
MRNUMBER = {2358379},
MRREVIEWER = {Dmitry Y. Kleinbock},
DOI = {10.1090/S0273-0979-07-01194-9},
ZBLNUMBER = {1194.11075},
}
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